Spread 1 tablespoon of cranberry sauce in center of 4 patties and top with 1 slice cheese; place remaining 4 patties on top, pinching edges together to seal.
Lightly coat large nonstick pan with cooking spray and place over medium heat.
Cook burgers for 4 minutes per side, or until well done. (Alternatively, cook on grill.)
Season with salt and pepper. Place on buns and top with cranberry mayo* and arugula.
For the Cranberry Mayo
In a small bowl, stir together cranberry sauce and mayonnaise.