In a medium bowl, combine ground beef and Worcestershire sauce. Shape into twelve half inch thick patties. Set aside.
Combine sauce ingredients in small saucepan and bring to a boil over medium heat, stirring frequently. Remove from heat.
Place pineapple slices on rack in broiler and brush with sauce. Put pan in oven so the pineapples are 3 to 4 inches from the heat. Broil 4 minutes, turning once and brushing with more sauce. Remove pineapple and keep warm.
Place burgers on a rack in broiler pan so surface of beef is 3-4 inches from the heat. Broil 7 to 8 minutes to medium (160°F) well, until beef is not pink in center and juices show no pink color. Turn once and brush burgers with remaining sauce after turning.
Cut each pineapple slice into thirds. Line bottom of each Hawaiian sweet roll with lettuce, top with a burger, then with a pineapple piece.
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