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 heat. Grill, covered but turning occasionally, 7 to 10 minutes until instant-read thermometer inserted horizontally into center registers 160°F.
Evenly spread goat cheese on bottom half of each biscuit; top with burgers. Evenly spoon peach compote over burgers. Close sandwiches.
