Combine ground beef, mushrooms, egg white, garlic, salt and pepper in large bowl, mixing lightly but thoroughly. Lightly shape into four 1/2-inch thick patties.
Place patties on grid over medium, ash-covered coals. Grill, covered, 8 to 10 minutes (over medium heat on preheated gas grill, covered, 7 to 9 minutes) until instant-read thermometer inserted horizontally into center registers 160°F, turning occasionally.
Place burgers on bun bottoms. Top evenly with cheese. Close sandwich.
Nutrition information per serving: 338 Calories; 117 kcal Calories from fat; 13 g Total Fat; 6 g Saturated Fat; 4 g Monounsaturated Fat; 85 mg Cholesterol; 644 mg Sodium; 23 g Total Carbohydrate; 1.5 g Dietary Fiber; 33 g Protein; 4.1 mg Iron; 9 mg Niacin; 0.4 mg Vitamin B6; 2.3 mcg Vitamin B12; 6.1 mg Zinc; 34.9 mcg Selenium; 88.4 mg Choline;