Arrange onion slices in a single layer in a 13- x 9-inch baking dish.
Stir together vinegar and next 4 ingredients. Pour over onion slices. Cover and chill 8 hours, turning once.
Remove onions from dressing with a slotted spatula, reserving dressing.
Grill onion slices, covered with grill lid, over medium-high heat (350° to 400°) 2 minutes on each side or until crisp-tender and golden.
Layer half of grilled onion slices each with 1 cheese round, 2 basil leaves, and 1 tomato slice; top with remaining half of onion slices.
Grill onion stacks, covered with grill lid, over medium-high heat 3 minutes or until cheese melts. Serve onion stacks with reserved dressing. Garnish, if desired.