1. Preheat the oven to 400˚. Line two rimmed baking sheets with parchment paper.
2. Unfold each puff pastry rectangle onto a prepared baking sheet and cut into 4 squares. Place on the baking sheets, spacing the squares at least 1 inch apart.
3. Use a sharp knife to score a border ¼ inch from the edges on each square. Do not cut all the way through the dough. Transfer the baking sheets to the freezer for 10 minutes.
4. In a medium bowl, mix together the tomatoes, olive oil, and thyme.
5. Make an egg wash by whisking 1 of the eggs with 1 tablespoon water in a small bowl.
6. Remove the baking sheets from the freezer and brush the borders of each square with the egg wash. Sprinkle the cheese evenly over each square, staying inside the border. Lay a piece of prosciutto over the cheese and place the tomatoes in a rough square just inside the border.
7. Bake for 18-20 minutes, until the pastries are golden brown and slightly puffed. Remove the baking sheets from the oven.
8. Carefully crack an egg into the center of each tart and sprinkle with salt and pepper. Return to the oven and bake for an additional 9-11 minutes, until the whites are set, and the yolks are still runny.
9. Remove from the oven and use a large metal spatula to transfer the tarts to plates. Serve immediately.