Preheat oven to 375°. Melt butter in a 12-inch ovenproof skillet over medium-low heat; swirl to coat bottom. Sprinkle with sugar.
Place 12 to 14 onion halves in skillet, cut sides down, with sides touching. Cut remaining onion halves into quarters, and place, cut sides down, in gaps between onions in skillet.
Cover and cook onions 10 minutes, adding salt and 1 Tbsp. water halfway through. Add vinegar; cover and cook 3 to 5 minutes or until onions are caramelized. Remove from heat.
Roll pastry sheet into a 12-inch square on a lightly floured surface; cut into a 12-inch round. Place pastry round over onions in skillet.
Bake at 375° for 30 minutes or until golden. Remove from oven, and immediately invert onto a serving plate.
Note: We tested with Dufour Pastry Kitchens Classic Puff Pastry.
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