When did Dumbledore find Harry engaging?

Question

In Chapter Thirty-Three of Deathly Hallows Snape gives Dumbledore his (negative) impression of Harry, and Dumbledore responds that he finds Harry to be an engaging child:

The office dissolved but re-formed instantly. Snape was pacing up and down in front of Dumbledore.

“— mediocre, arrogant as his father, a determined rule-breaker, delighted to find himself famous, attention-seeking and impertinent —”

“You see what you expect to see, Severus,” said Dumbledore, without raising his eyes from a copy of Transfiguration Today. “Other teachers report that the boy is modest, likable, and reasonably talented. Personally, I find him an engaging child.”

What did Dumbledore base this assessment on? This conversation is clearly taking place towards the beginning of the term in Harry’s first year, possibly even after his very first class with Snape. At this point in the story there have been zero recorded interactions between Dumbledore and Harry. The first time we see them interact is after Christmas in the classroom with the Mirror of Erised.

Of course, one could posit that the books don’t show every moment during the year and they may have interacted off-camera. However, this doesn’t seem so likely, as when they do meet at the Mirror there doesn’t seem to be any particular familiarity between them. And we see throughout the series that Dumbledore remains pretty aloof from the students (excepting the special relationship he later cultivates with Harry). Even months after the meeting at the Mirror, Harry doesn’t seem to have any additional familiarity with Dumbledore:

“We’ve got to go to Dumbledore,” said Harry. “Hagrid told that stranger how to get past Fluffy, and it was either Snape or Voldemort under that cloak — it must’ve been easy, once he’d got Hagrid drunk. I just hope Dumbledore believes us. Firenze might back us up if Bane doesn’t stop him. Where’s Dumbledore’s office?”

They looked around, as if hoping to see a sign pointing them in the right direction. They had never been told where Dumbledore lived, nor did they know anyone who had been sent to see him.

Even six years later we see that students other than Harry (once Dumbledore took an active interest) had little or nothing to do with Dumbledore:

This was stretching the truth to breaking point; as far as Harry knew, Ron and Dumbledore had never been alone together, and direct contact between them had been negligible.

So when did Dumbledore actually form this impression of Harry?

Answer 1

Dumbledore's statement

I find him an engaging child

does not necessarily require him to have interacted with Harry directly. It's quite possible that Dumbledore observed Harry, possibly without the latter's knowledge, in order to form this opinion.

There are a number of places that might suggest that Dumbledore has been keeping an eye on Harry beyond their direct interactions. For example, Dumbledore's statement at the end of Order of the Pheonix:

I defy anyone who has watched you as I have — and I have watched you more closely than you can have imagined — not to want to save you more pain than you had already suffered.

(emphasis mine)

Of course, Dumbledore might have been invisible while performing his various investigations; he certainly was during at least one instance, when watching Harry and Ron at the Mirror of Erised:

"And it showed your friend Ron himself as head boy."

"How did you know —?"

"I don't need a cloak to become invisible," said Dumbledore gently.

In short, we don't know exactly when Dumbledore formed this opinion. He might even have been watching Harry long before he arrived at Hogwarts. The point, though, is that there could have been many opportunities for him to collect observations to base his assessment on, even if Harry was not aware of them.

Answer 2

Are we forgetting he knew Lily and James and was familiar with Harry before Harry even realized who he was? Lily probably sent a letter to Dumbledore like she did to Sirius.

Answer 3

There is no indication regarding when this interaction takes place, other than the last line:

Dumbledore turned a page, and said, without looking up, “Keep an eye on Quirrell, won’t you?”

So we know it takes place sometime between when Harry started learning in Hogwarts, and the first book's finale. Because Dumbledore says that he finds Harry "engaging", meaning he already spoke with Harry, the most probable conclusion is that it takes place after their first interaction with the mirror of Erised, a few months after school started.