Combine Ground Beef and Worcestershire sauce in medium bowl, mixing lightly but thoroughly. Lightly shape into twelve 1/2-inch thick mini patties. Set aside.
Combine sauce ingredients in small saucepan; bring to a boil over medium heat, stirring frequently. Remove from heat.
Place patties on grid over medium, ash-covered coals. Grill, covered, 8 to 10 minutes (over medium heat on preheated gas grill, covered, 9 to 11 minutes) until instant-read thermometer inserted horizontally into center registers 160°F, turning occasionally.
Meanwhile brush pineapple slices with sauce and place on grid around patties. Grill pineapple 4 minutes, turning once and brushing with additional sauce. Remove pineapple, keep warm. Brush burgers with remaining sauce after turning.
Cut each pineapple slice into thirds. Line bottom of each roll with lettuce, top with burger, then with pineapple piece. Close sandwiches.
Nutrition information per serving: 582 Calories; 117 kcal Calories from fat; 13 g Total Fat; 4 g Saturated Fat; 5 g Monounsaturated Fat; 117 mg Cholesterol; 807 mg Sodium; 81 g Total Carbohydrate; 4.6 g Dietary Fiber; 33 g Protein; 6.7 mg Iron; 8.7 mg NE Niacin; 0.4 mg Vitamin B6; 2.4 mcg Vitamin B12; 6.7 mg Zinc; 49.8 mcg Selenium;