Fontina cheese adds a lot of flavor, but if you can't find it, use whole-milk mozzarella.
Preheat oven to 500 degrees. Melt butter in a medium skillet; saute leek over medium-high heat, stirring occasionally, until tender and starting to color, 3 to 4 minutes. Add mushrooms; saute until tender, about 2 minutes. Season with 1/4 teaspoon salt and a pinch of pepper.
Brush a 12-inch cast-iron pan with 1 teaspoon oil. Press dough flat in skillet, spreading to edge (if it retracts, let rest 5 minutes before continuing). Brush a 1-inch border around edge with another teaspoon oil. Sprinkle 1 cup fontina over dough. Top with leek mixture and remaining cup cheese.
Cook over medium-high heat until bottom of crust is golden brown, 4 to 5 minutes. Transfer to oven, and bake until edge of crust is golden brown and dough is cooked through, about 10 minutes. Drape prosciutto over pizza.