Preheat the oven to 400 degrees F.
Butter a baking dish and set aside. On a pasta board, or large, clean, wooden cutting board, make a mound out of flour and make well in center, just like making fresh pasta. This can also be done in a large mixing bowl. Place the sugar, eggs, and melted butter into the well. Using a fork, mix the ingredients, bringing the walls of the flour into the well bit by bit, and forming a mass of dough.
Once the dough has come together, knead it for 1 minute until a medium ball has formed. Divide the dough into 4 equal pieces and roll each into a ball. Wrap each dough ball in plastic. Refrigerate the balls for 15 minutes so that the dough has a chance to rest.
Meanwhile, cut the figs into halves. Remove the pastry from the refrigerator and roll each piece into a 4 to 5-inch circle, about 1/4-inch thick. Carefully place the rounds in the prepared baking sheet. Arrange the fruit cut side up in the center of each pastry round. If you desire, you can fold the tart edges in over the fruit to form a kind of shell, or else let the fruit come near the flat edges of the tarts, forming a flat, disk-like tart. Bake the tarts until the pastry is golden brown, about 20 to 22 minutes.
Remove the tarts from oven, lay 2 thin slices of prosciutto over each, and sprinkle them with cracked black pepper, fennel seeds and the extra-virgin olive oil. Serve immediately.