Heat grill or grill pan to medium-high. Clean and lightly oil hot grill. Brush chiles and onion with oil, then season with salt and pepper. Grill vegetables, turning occasionally, until crisp-tender, 10 minutes. Transfer to a cutting board. Slice chiles, discarding stems and seeds.
Season patties with salt and pepper, then brush with oil. Grill until cooked through, about 4 minutes per side; during last minute of cooking, top each burger with cheese and cover grill. Serve burgers on buns with chiles, onion, and other desired toppings.