Following from the Basilisk petrification query, perhaps it's not actually the eyes per se that do the killing? Maybe your basilisks have some other kind of terminal mechanism in their arsenal?
Pliny says that basilisks are small and can kill shrubberies and split rocks and that only the weasel is impervious to their venom. (Smithsonian) Others have depicted the creature as being rather larger; some are serpentine, some are birdlike, one depiction looks like a ten legged chicken-snake. But anyway...on to death!
Perhaps the basilisks in your world can sing a deadly song of infrasound. According to that article, infrasound can cause, among several cacatorially amusing symptoms, rather more severe symptoms such as organ rupture and death.
I'd suggest your basilisks, apart from being immune to their own infrasound capabilities, are able to home in on the resonant frequency at which its victim will react, and then ramp up the wattage until the power of the sound itself causes its victim to collapse into a heap of disintegral ooze. The powerful infrasound of the basilisk simply shakes them to pieces, in a sense. Because the sound must be focused, it simply appears to onlookers & survivors that the basilisk is staring them to death.