In a bowl, combine giardiniera and sugar; set aside., Place cheese in buns; top with sausages and giardiniera mixture. Wrap individually in a double thickness of heavy-duty foil (about 12x10 in.). Grill, uncovered, over medium heat for 8-10 minutes or until heated through and cheese is melted. Open foil carefully to allow steam to escape. If desired, serve with additional giardiniera.