Preheat the oven to its highest setting (500 degrees F). Preheat a cast-iron grill pan to medium heat.
Melt the butter in a large skillet over medium-high heat. Add the leeks and saute for about 4 minutes. Add the mushrooms and saute for another 2 minutes. Season with salt and pepper. Set aside.
Lightly flour a cutting board and roll out the pizza dough into an oval shape that will fit the grill pan. Coat the preheated grill pan with 1 tablespoon olive oil. Brush the dough with 2 teaspoons olive oil and place it, oiled-side down, on the grill pan¿the dough will start to bubble as it cooks. Cook until grill marks form, 3 to 5 minutes; flip, and continue cooking on the reverse side until grill marks form and the dough has stiffened, about 3 minutes.
Remove the pizza to a baking sheet, top with the mushrooms-leek mixture and sprinkle with the cheese. Place in the oven and cook until the cheeses melts, 5 to 7 minutes depending on how hot your oven is.
Remove the pizza from the oven and add the prosciutto. Slice into small bites and serve.