Tomato Tarte Tatin

Preheat oven to 300°F. Place sugar and vinegar in a 10-inch ovenproof stainless steel skillet; heat over low, swirling and tilting skillet occasionally, until mixture is light golden brown, 5 to 6 minutes. Increase heat to medium-low; add 1 tablespoon of the butter, and swirl skillet to spread. Stir in onion, thyme, and salt; cook, stirring often, until onion is lightly caramelized and sauce is deep golden, 4 to 5 minutes. Remove skillet from heat; arrange tomatoes, cut sides down, on top of onion slices.

Place skillet in preheated oven, and bake until tomato skins are wrinkly and have released juices, about 1 hour, 10 minutes. Remove skillet from oven. Transfer tomatoes from skillet using a slotted spoon to a paper towel-lined baking sheet. Transfer onion slices to a paper towel-lined plate; discard remaining juice. Increase oven temperature to 425°F.

While oven is preheating, coat a 9-inch springform pan with remaining 1 tablespoon butter. Gently pat roasted tomatoes using a paper towel to remove extra moisture; arrange tomatoes, cut sides up, in an even layer in prepared springform pan until all tomato halves are used. Pat onion mixture dry using a paper towel; place onion slices over tomatoes. Top with grated Parmesan cheese. Unfold puff pastry sheet onto a lightly floured work surface. Gently roll into a 10-inch square, and cut into a 10-inch round. Carefully center and place round pastry over onion slices, tucking in edges. Prick pastry using the tines of a fork in a handful of spots.

Bake at 425°F until crust is golden brown, 25 to 30 minutes. Transfer springform pan to a wire rack; let cool for 5 minutes. Run a sharp knife around edges to loosen; remove springform pan ring from pan. Center a plate over pastry crust; carefully invert tarte tatin onto plate. Garnish with additional Parmesan and thyme. Slice and serve immediately.