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BEAT egg and milk with wire whisk in medium saucepan until well blended. Add granulated sugar and tapioca; stir until well blended. Let stand 5 min.
BRING to full rolling boil on medium heat, stirring constantly. Remove from heat; stir in vanilla. Pour into shallow 1-qt. baking dish. Cover with plastic wrap, placing wrap directly on surface of pudding. Refrigerate 2 hours or until completely cooled. (Pudding thickens as it cools.)
PREHEAT broiler just before serving. Uncover pudding. Arrange banana slices in single layer over pudding. Sprinkle evenly with brown sugar. Broil, 3 inches from heat, 2 to 3 min, or until sugar is melted and caramelized.