Make salsa: Preheat oven to 400°. Line a baking sheet with foil and top with tomatoes, onion, garlic, and jalapeño. Bake until softened, about 25 minutes, turning ingredients halfway through baking. Remove from oven and let cool.
Core tomatoes, peel onion and garlic, and stem jalapeño. Put the vegetables, lime juice, cilantro, and salt in a food processor and pulse just until chunky and incorporated. Transfer salsa to a bowl.
Prepare chiles: Preheat broiler. Arrange poblanos on foil-lined baking sheet in a single layer and broil 2 to 3 in. from heat until blackened, turning as needed, 10 to 12 minutes. Put chiles in a bowl, cover, and let sit 10 minutes to loosen skins.
Peel chiles, pull out stems (keeping chiles as intact as possible), and remove seeds with a spoon or your fingers. Blot with paper towels. Preheat oven to 200°.
In a bowl, combine chopped onion and shredded cheese. Squeeze handfuls to form 6 narrow logs. Carefully stuff logs into chiles through stem ends.
Pour oil into a large pot to a depth of 2 to 3 in. and heat to 375°. Meanwhile, separate 5 eggs. In the bowl of a mixer, beat whites until soft peaks form. In another bowl, whisk yolks with 1/3 cup flour and 1/4 tsp. salt. Add a spoonful of whites to yolk mixture and whisk to combine, then gently fold yolk mixture into whites just until incorporated.
Put 1/2 cup flour in a shallow bowl. Using tongs or your hands, dip a chile first into flour, then into egg batter, turning gently to coat. Gently lower chile into hot oil. Repeat with a second chile. Cook until browned, gently turning once with tongs, about 4 minutes total. Transfer chiles to a baking sheet lined with paper towels and keep warm in the oven. Cook remaining chiles the same way.
To poach eggs, fill a wide, deep frying pan with enough water to cover an egg out of the shell by about 1 in. Heat water until 1 or 2 bubbles break the surface. Reduce heat until bubbles form on pan bottom and occasionally pop to the surface. Break open remaining 6 eggs, one at a time, into water, keeping eggs close to the surface. Cook until eggs are done the way you like, 3 to 5 minutes for soft yolks with firm whites or 7 to 10 minutes for firm yolks and whites. Remove eggs from water with a slotted spoon and transfer to a plate.
Divide chiles among 6 plates, top each with a poached egg and a spoonful of salsa, and serve remaining salsa on the side.
Make ahead: Salsa keeps 1 day, covered and chilled.
Note: Nutritional analysis is per serving.