To prepare barbecue sauce, combine brown sugar, mustard, beer and vinegar in small saucepan; bring to a boil. Reduce heat and simmer 15 to 17 minutes until thickened and reduced in to 1/2 cup, stirring occasionally.
Lightly shape Ground Beef 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. About 2 minutes before burgers are done, place bun, cut sides down, on grid. Grill until lightly toasted.
Spoon 1 tablespoon sauce on bottom of each bun; top with burger. Evenly spoon remaining sauce on burger. Evenly top burgers with cabbage. Close sandwiches.
Nutrition information per serving: 370 Calories; 92.7 kcal Calories from fat; 10.3 g Total Fat; 3.7 g Saturated Fat; 0.2 g Trans Fat; 1.2 g Polyunsaturated Fat; 3.9 g Monounsaturated Fat; 84 mg Cholesterol; 463 mg Sodium; 38 g Total Carbohydrate; 1 g Dietary Fiber; 17 g Total Sugars; 31 g Protein; 13 g Added Sugars; 102.9 mg Calcium; 4.8 mg Iron; 500 mg Potassium; 0 mcg Vitamin D; 0.3 mg Riboflavin; 8.2 mg NE Niacin; 0.5 mg Vitamin B6; 2.9 mcg Vitamin B12; 288 mg Phosphorus; 6.9 mg Zinc; 38.2 mcg Selenium; 107.4 mg Choline;