Gently combine first 4 ingredients. Shape mixture into 6 thin patties; cover and chill 30 minutes.
Melt 1 Tbsp. butter in a nonstick skillet over medium-high heat; add onion and rosemary, and sauté 10 minutes or until onion is tender and golden. Remove onion mixture from skillet.
Melt remaining 1/2 Tbsp. butter in same skillet over medium heat. Cook patties 6 minutes on each side or until no longer pink in center. Place on an aluminum foil-lined baking sheet or broiler pan coated with cooking spray.
Top each patty evenly with onion mixture and 1 cheese slice. Broil 6 inches from heat 3 to 5 minutes or until cheese is lightly browned. Serve on hamburger buns with desired toppings.
*4 oz. blue cheese, Gorgonzola cheese, or Swiss cheese may be substituted.
Note: Nutritional analysis does not include toppings.