Preheat the oven to 400 degrees. Line 2 large baking sheets with parchment paper. Roll out each puff pastry sheet to a 12 x 14-inch rectangle. Place on the baking sheets and fold over the edges of each pastry to form a 1/2-inch border. Prick the bottom of each pastry with a fork and bake for 10 minutes. Let cool for 15 to 20 minutes.
Meanwhile, slice the tomatoes in half widthwise and gently squeeze out the seeds and juice. Slice the tomatoes into 1/8-inch-thick rounds.
Using a rubber spatula, spread half of the fromage blanc over one of the cooled crusts. Cover the fromage blanc with half of the tomatoes, slices overlapping, then season with salt. Top the tomatoes with half of the onion slices, dot with half of the olives and sprinkle with half of the thyme and some pepper.
Bake the tart for 18 minutes, or until the tomatoes are very soft but before the onion gets too brown.
While the first tart is baking, assemble the other tart with the remaining ingredients. Bake the second tart when the first one comes out of the oven. (It's better not to bake the tarts at the same time.).
Right before serving, drizzle the tarts with the olive oil and sprinkle with basil.