Preheat the oven to 425° and line a large baking sheet with parchment paper. Position racks in the middle and upper thirds of the oven. On a lightly floured surface, roll out the puff pastry to a 9 1/2-by-17 1/2-inch rectangle. Using a straight edge, trim the pastry to a 9-by-17-inch rectangle. Transfer the pastry to the baking sheet and poke all over with a fork. Top with another sheet of parchment and another baking sheet and bake for 25 minutes on the middle rack, until golden. Remove the top sheet and parchment paper and bake the pastry until lightly browned and dry, about 10 minutes longer. Slide the paper and pastry onto a rack and let cool.
Meanwhile, in a large bowl, toss the tomatoes with the olive oil and 2 teaspoons of thyme and season with salt and pepper. Place the tomatoes on a baking sheet, cut side up, and bake on the upper rack for about 15 minutes, until softened slightly. Let cool.