To prepare peach compote, heat large skillet over medium heat until hot. Add peaches and onion; cook and stir until lightly browned about 5 minutes. Add beer, brown sugar and rosemary; bring to a boil. Cook until peaches and onion are soft and liquid is almost gone, 13 to 15 minutes; keep warm.
Lightly shape ground beef into four 1/2-inch 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.
Evenly spread goat cheese on bottom half of each biscuit; top with burgers. Evenly spoon peach compote over burgers. Close sandwiches.
Nutrition information per serving: 475 Calories; 180 kcal Calories from fat; 20 g Total Fat; 8 g Saturated Fat; 4 g Monounsaturated Fat; 81 mg Cholesterol; 704 mg Sodium; 42 g Total Carbohydrate; 2 g Dietary Fiber; 30 g Protein; 4.3 mg Iron; 5.4 mg Niacin; 0.4 mg Vitamin B6; 2.2 mcg Vitamin B12; 5.7 mg Zinc; 19 mcg Selenium; 79.3 mg Choline;