Preheat oven to 425 degrees F. Combine butter and olive oil in 3-quart rectangular baking dish. Whisk in vinegar, wine (or broth or water), sugar, salt and pepper. Set aside.
Peel off papery outer layers of onions, but do not cut off either end. Cut onions in half from stem through root end. Place onions in dish, cut-sides up. Cover loosely with foil and bake for 30 minutes.
Remove foil. Using tongs, carefully turn onions over to cut-sides down. Bake, uncovered, for 20 to 25 minutes longer or until onions are tender and balsamic mixture is thickened and caramelized. Serve cut-sides up. If desired, sprinkle with fresh thyme.
To make ahead: Prepare onions as directed. Cool, cover, and chill overnight. To reheat, preheat oven to 325 degrees F. Bake, covered, about 40 minutes or until warm.
1 fat, 1/2 other carbohydrate, 1/2 vegetable