The five-letter word we're looking for is
SOTOL, which is appropriately a type of liquor.
To get there, we must first
figure out the names of all the cocktails on the menu. Using the partially-censored list of ingredients plus the enumerations, we can come up with this list:
BOURBON + SUGAR SYRUP + WATER + CRUSHED ICE + MINT LEAVES (4,5) - MINT JULEP
BRANDY + RUBY PORT + EGG (5,4) - PORTO FLIP
CAMPARI + GIN + VERMOUTH ROSSO + ORANGE PEEL (7) - NEGRONI
CHAMPAGNE + ORANGE JUICE (6) - MIMOSA
CLUB-MATE + GOLD RUM + LIMES + BROWN SUGAR + ICE (7) - TSCHUNK
COGNAC + ORANGE LIQUEUR + FRESH LEMON JUICE (7) - SIDECAR
CREME DE CASSIS + WHITE WINE (3) - KIR
CREME DE MURE + DRY GIN + FRESH LEMON JUICE + SUGAR SYRUP + ICE (7) - BRAMBLE
DARK RUM + GINGER BEER + ICE + LIME SLICE (4,1,6) - DARK 'N' STORMY
DRY GIN + CHERRY BRANDY LIQUEUR + BENEDICTINE + TRIPLE SEC + PINEAPPLE JUICE + FRESH LIME JUICE + GRENADINE + ANGOSTURA BITTERS (9,5) - SINGAPORE SLING
ESPADIN MEZCAL + FALERNUM + JAMAICAN OVERPROOF WHITE RUM + MARASCHINO LUXARDO + FRESH LIME JUICE + SIMPLE SYRUP + EGG WHITE (7) - ILLEGAL
GIN + DRY VERMOUTH + MARASCHINO + ABSINTHE + ORANGE BITTERS (6) - TUXEDO
GIN + PEACH BITTERS + MINT LEAVES (5) - DERBY
GRAPPA + HONEY SYRUP + FRESH LEMON JUICE (5) - VE.N.TO
PISCO + FRESH LIME JUICE + SIMPLE SYRUP + EGG WHITE + ANGOSTURA BITTERS (5,4) - PISCO SOUR
PROSECCO + DIGESTIVE BITTERS + SODA WATER (6) - SPRITZ
RED WINE + COLA (9) - KALIMOTXO
RUM + COCONUT MILK + CREAM OF COCONUT + SWEETENED CONDENSED MILK + VANILLA + NUTMEG + CLOVE + CINNAMON (7) - COQUITO
RUM + CREAM OF COCONUT + PINEAPPLE JUICE (4,6) - PINA COLADA
RYE WHISKEY + FERNET-BRANCA + ANGOSTURA BITTERS + SIMPLE SYRUP (7) - TORONTO
SUGAR + BITTERS + WATER + WHISKEY + ORANGE ZEST + COCKTAIL CHERRY (3,9) - OLD FASHIONED
TEQUILA + FRESH LIME JUICE + GRAPEFRUIT SODA (6) - PALOMA
VODKA + GINGER BEER + FRESH LEMON JUICE + LIME WEDGE + MINT (6,4) - MOSCOW MULE
VODKA + KAHLUA + ICE (5,7) - BLACK RUSSIAN
VODKA + PEACH SCHNAPPS + ORANGE JUICE + CRANBERRY JUICE (3,2,3,5) - SEX ON THE BEACH
WHITE RUM + SUGAR + FRESH LIME JUICE + SODA WATER + MINT (6) - MOJITO
As for the flags, those represent
the countries of origin (or, in the case of quite a few of them, the presumed or contested countries of origin) of each cocktail, listed from left-to-right in alphabetical order of the name of the cocktail - e.g. the BLACK RUSSIAN, first on the list, was invented in Belgium, while the VE.N.TO, last in the order, is Italian. (In case you're curious, the one with three flags is the SIDECAR, which has been variously attributed to bartenders in Paris, London, and New York City.)
So now we have our menu. Now what to do with the graphic?
Well, we have 26 cocktails and 26 regions in the graphic, so we must need to fill each region with a different cocktail. Some have unique lengths and can be filled in immediately, but how do we disambiguate the remaining cocktails? Using the champagne bubbles, of course! Each region contains a single circle, and we're told that each champagne bubble is unique, so we must have to put a different letter in each circle. This lets us place other names - e.g. COQUITO is the only cocktail name with a Q in it, and there is only one 7-letter region with a bubble at the third position - and using a combination of letter frequencies and logic, we can fill in the grid as follows:
A full explanation of the logic used to fill the grid can be seen below.
Finally, to extract the answer,
note the italicized phrase "what they most frequently see selected" near the end of the puzzle. We're looking for a 5-letter word, and the grid is conveniently divided into 5 vertical regions by color along the top and bottom. If we look at each section and determine which letter occurs most frequently in that section, we should be able to get an answer!
It turns out
that the most common letters, from left to right, are S, O, T, O again, and L, which spell SOTOL!
Full grid explanation:
First, as mentioned above, there are some names with unique lengths, which forces their position in the grid:
Then, there are a few letters which only appear once in the list of names, so we know those letters must be highlighted and can fill in the cocktails accordingly (note that in order for the puzzle to have a single solution, the grid has been set up so that each region of the same length has its bubble in a different position):
This has allowed us to differentiate a couple of length pairs (MOSCOW MULE and PINA COLADA are the only cocktails of length 10, and similarly with DERBY and VE.N.TO for length 5) so we can fill those in as well:
Now, we'll need to start using both lengths and champagne bubbles to differentiate answers. Let's start with our final length pair, at 12 letters long: OLD FASHIONED and BLACK RUSSIAN. The two 12-length regions (starting below the D and Y in DARK 'N' STORMY) have their bubbles at the 4th and 7th positions. (From here, I'll be denoting regions using the notation [length, bubble position], so these two are [12,4] and [12,7].) BLACK RUSSIAN must therefore highlight a C or a U, while OLD FASHIONED must highlight an F or an H. However, H has already been claimed by SEX ON THE BEACH, so we must put OLD FASHIONED in [12,4] and BLACK RUSSIAN in [12,7]:
Now things get interesting. With four words of length 9, seven of length 7 (!) and five of length 6, we simply have too many options to be able to effectively disambiguate which can go where. Instead, let's look at some rare letters. X is certainly one of the rarer letters which we have left; while it appears three times in our answer list, one (SEX ON THE BEACH) is already on the board, so we're left with KALIMOTXO and TUXEDO. It can't be KALIMOTXO, however, as there is no [9,8]. Therefore, we must put TUXEDO in the [6,3], just to the left of the glass's stem. We can also look at J, which is shared only by MINT JULEP and MOJITO. By complete coincidence, we've just filled in the [6,3], which rules out MOJITO, so MINT JULEP must go in the [9,5]. Surprisingly, the other letters we can look at are G and B. Our only remaining Gs are in NEGRONI and ILLEGAL, but note that we already have COQUITO in the [7,3], and which would conflict with using NEGRONI for G. Thus, G must be highlighted in ILLEGAL, which will go in the [7,5]. Meanwhile, the only Bs we have left are both in BRAMBLE, and since (by complete coincidence ;) ) we've just filled in [7,5], it must be the first B, placing BRAMBLE in [7,1]. Let's take a look at the grid now:
We're still not quite done with rare letters. Let's look at M. Of our remaining cocktails, only KALIMOTXO, MIMOSA, PALOMA, and MOJITO contain an M. Our [9,5] has already been filled, so we can rule out KALIMOTXO and look only at the remaining three six-letter cocktails. We do have both a [6,1] and a [6,5], so we can't immediately rule any out, but what would happen if we put PALOMA in the [6,1]? Well, obviously PALOMA couldn't feature the M anymore, but neither could MIMOSA or MOJITO, as the [6,1] and [6,3] would already be used. Therefore, either MIMOSA or MOJITO must go in the [6,1], meaning the M must be highlighted in that bubble, ruling out PALOMA from going in the [6,5]. Thus, it must go in [6,2]. With PALOMA ruled out, we can figure out P as well: we're left with PORTO FLIP and PISCO SOUR, and with no [9,1], we must put PORTO FLIP in the [9,9]:
Now, we only have three Cs left to fill in: one in PISCO SOUR, one in TSCHUNK, and one in SIDECAR. However, we can rule out both 7-letter answers because we've already used both the [7,3] and the [7,5], so we must put PISCO SOUR in the [9,4]. This leaves us with only one nine-letter answer remaining, so we can put in KALIMOTXO along the bottom. That uses up our T, meaning MOJITO can't occupy the [6,5], so it goes in the [6,1] in the stem while MIMOSA fills in immediately alongside it:
Now, we have four remaining seven-letter words and four letters remaining: E, I, N, and O. TSCHUNK only contains one of those letters, so it must fill into the [7,6] in the top right. TORONTO also contains only one of those letters, but amazingly, its three Os also happen to align perfectly with the three spaces we have remaining ([7,2], [7,4], and [7,7]). That leaves NEGRONI and SIDECAR to be the E and I in some order. NEGRONI could be [7,2] and feature the E or it could be [7,7] and feature the I, while SIDECAR could be [7,2] and feature the I or [7,4] and feature the E! What to do? Well, we have to notice that if NEGRONI is in the [7,2], that means SIDECAR has to be in [7,4], and both bubbles will have Es in them. Therefore, NEGRONI must be in [7,7], SIDECAR must be in [7,4], and TORONTO fills in the [7,2] to complete the grid:
