Preheat a grill to medium high. Grill the serrano, Cubanelle and poblano peppers, turning, until charred all over, 6 minutes for the serrano, 8 minutes for the Cubanelle, and 10 to 12 minutes for the poblano. Transfer to a large bowl, cover with plastic wrap and let steam 10 minutes. Peel, seed and chop all the peppers.
Heat the vegetable oil in a medium saucepan over medium-high heat. Add the onion, season with salt and cook, stirring, until browned, 5 to 6 minutes. Add the garlic and cumin and cook, stirring, until the cumin is toasted, about 1 minute. Add 1 cup water, the grilled peppers and 1/2 teaspoon salt. Simmer over medium heat until most of the liquid has cooked off, 8 to 10 minutes. Stir in the vinegar and season with salt.
Form the ground beef into 8 patties, about 4 inches wide and 1/2 inch thick; season with salt and pepper. Oil the grill grates. Grill the burgers over medium-high heat, 3 to 31/2 minutes per side for medium rare; top each with a slice of cheese after flipping and cover to melt. Toast the buns cut-side down, about 30 seconds.
Spread some mayonnaise and barbecue sauce on the buns. Serve the patties on the buns with lettuce, sliced onion and the grilled chile salsa.