Make 2 equal sized patties out of each round so you have 8 patties total.
Sprinkle 4 of the patties with 2 tablespoons red peppers and cheese, and top with remaining patties, working the turkey around the edges to seal burgers closed.
Season with salt and a few grinds of pepper.
Grill or broil until cooked through, about 5 minutes per side.