Grill the eggplants. The easiest way is to place them directly on the gas stove. Move and rotate them occasionally for even charring until cooked through. Cool the eggplants.
While the eggplants cool, peel and mince the garlic; peel and thinly slice the shallots; and roughly chop the tomatoes.
When the eggplants are cool enough to handle, use paper towels to rub off the charred skins. Finely chop the eggplants.
Heat the cooking oil in a wok or frying pan. Saute the garlic, shallots and tomatoes for about a minute. Add the chopped eggplants. Season with fish sauce. Continue sauteeing until the mixture appears dry.
Pour the beaten eggs in a thin stream, stirring the contents of the pan with one hand as your pour the eggs with the other. Drizzle in more fish sauce. Cook, stirring, just until the eggs are set.
Serve the poqui-poqui at once.