Preheat your oven to 375°F (190°C). Slice the tomatoes into thick rounds, about 1/2 inch thick. Pat them dry with a paper towel to remove excess moisture.

In a 10-inch oven-safe skillet, melt the butter over medium heat. Add the sugar and stir until it dissolves and begins to caramelize. Arrange the tomato slices in a circular pattern in the skillet, slightly overlapping them. Sprinkle with thyme, salt, and black pepper. Drizzle with balsamic vinegar and olive oil. Cook for 5-7 minutes until the tomatoes soften and the caramel deepens in color.

Roll out the puff pastry sheet to fit the size of your skillet. Place it over the caramelized tomatoes, tucking the edges around the tomatoes. Prick the pastry with a fork to allow steam to escape.

Transfer the skillet to the preheated oven and bake for 25-30 minutes, or until the pastry is golden brown and crisp. Remove from the oven and let it cool for 5 minutes.

Carefully invert the tarte tatin onto a serving plate by placing the plate over the skillet and flipping it quickly. Serve warm, garnished with additional thyme if desired.