Divide turkey into 4 equal sized rounds. 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.