1. Preheat broiler to high.
2. To prepare relish, place first 4 ingredients on a foil-lined baking sheet. Coat vegetables with cooking spray. Broil 14 minutes, turning vegetables once. Wrap Cubanelle and serrano in foil; let stand 10 minutes. Peel and cut into 1-inch strips; discard peels and seeds. Combine tomatillo, Cubanelle, serrano, garlic, cilantro, and next 5 ingredients (through 1/4 teaspoon salt) in a mini food processor; pulse until combined.
3. To prepare burgers, brush onion slices with 1 1/2 teaspoons oil. Arrange onion in a single layer on a foil-lined baking sheet. Broil 6 minutes or until lightly charred, turning once.
4. Combine mushrooms, 2 garlic cloves, and black beans in a food processor; pulse until mixture is coarsely chopped. Add panko and next 5 ingredients (through egg); pulse until combined. Divide mixture into 6 equal portions; shape each portion into a 1/2-inch-thick patty.
5. Heat a large cast-iron skillet over medium heat. Add remaining 1 tablespoon olive oil to pan; swirl to coat. Add patties; cook 12 minutes, turning once. Add cheese to patties during last 2 minutes of cooking; cover pan to melt cheese. Place 1 lettuce leaf on bottom half of each bun. Top each lettuce leaf with 1 patty, 1 charred onion slice, and about 1 1/2 tablespoons relish. Top with top halves of buns.