Using your fingers, toss shallot, vinegar, sugar, ¾ tsp. salt, and 2 Tbsp. oil in a small bowl to combine. Let sit while you prepare the chiles.
Heat broiler. Arrange chiles on a small rimmed baking sheet. Drizzle all over with remaining 1 Tbsp. oil; season with salt and black pepper. Turn chiles cut side down and broil until slightly softened and skins are blistered all over, about 5 minutes.
Remove from oven and, using tongs, turn chiles cut side up. Divide cheese evenly among chile cavities. Finely grate garlic over, frequently tapping grater against baking sheet to help release garlic somewhat evenly over everything.
Return chiles to broiler and broil until cheese is melted and browned in spots, about 3 minutes.
Transfer chiles to a platter. Spoon shallot dressing over and season with more black pepper.