Broil chiles on a rimmed baking sheet, turning as needed, until blackened on all sides, about 10 minutes. Let cool. Peel, then cut open and remove stem and seeds. Finely chop chiles. Put in a medium bowl and stir in 1/4 tsp. salt. Mix mayonnaise with garlic in another medium bowl.
Heat a grill or griddle to medium (350° to 450°). Mix beef with remaining 1/2 tsp. salt in a large bowl until just combined. Divide into 4 portions and form each into a 1/2-in.-thick patty with a slight depression in the center.
Oil cooking grate or griddle with a wad of oiled paper towels. Grill burgers, turning once and sprinkling with about 1/2 tsp. soy sauce on each side, until done the way you like, 7 to 8 minutes for medium. During the last minute, lay a cheese slice on each burger to melt it.
Meanwhile, lightly butter cut sides of rolls and toast on grill or griddle, about 2 minutes.
Spread garlic mayo on cut sides of rolls. Add burgers, topping each with about 2 tbsp. chile mixture.