Combine Ground beef, salt and pepper in large bowl, mixing lightly but thoroughly. Lightly shape into eight thin mini-patties. Heat large nonstick skillet over medium heat until hot. Add patties; cook 5 to 6 minutes or until instant-read thermometer inserted horizontally into center registers 160°F, turning once. Top each burger with half cheese slice. Remove burgers; keep warm.Cook's Tip:Cooking times are for fresh or thoroughly thawed ground beef. Ground beef should be cooked to an internal temperature of 160°F. Color is not a reliable indicator of ground beef doneness.
Add onions and mushrooms to same skillet and cook for 2 to 4 minutes until tender. Add eggs and cook and stir 2 minutes until desired doneness. Season as desired.
Place burgers on bottoms of buns. Top burgers evenly with egg mixture. Close sandwiches.