Prepare the grill for direct cooking over medium heat (350° to 450°F).
Brush the cooking grates clean. Grill the pepper over direct medium heat, with the lid closed as much as possible, until the skin is blackened and blistered all over, 12 to 15 minutes, turning occasionally. Place the pepper in a small bowl and cover with plastic wrap to trap the steam. Set aside for at least 10 minutes, then peel the skin from the pepper, discard the stem and seeds, and cut into thin strips. Set aside.
In a small bowl gently combine the patty ingredients. Shape into four patties of equal size and thickness. Grill over direct medium heat, with the lid closed as much as possible, until the juices run clear, about 10 minutes, turning once. During the last 30 seconds of grilling time, lightly toast the buns over direct heat.
Serve the burgers hot on the toasted buns and top with grilled peppers, cheese, and lettuce.
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