Cut the onion into very thin slices. Put the vegetable oil and onions into a large skillet and place it over medium-low heat. Cook, stirring frequently, until the onions become golden brown and caramelized, about 30 minutes.
Place a pizza stone onto the floor of the oven and preheat the oven to 450 degrees F.
Remove the bones from the duck legs and shred the meat; set aside. Place the dough on a floured surface. Flatten it slightly, sprinkle with more flour, and roll it into a 10-inch disk. Sprinkle a wooden pizza paddle with flour and transfer the dough to the paddle. Shake the board a little to make sure the dough doesn't stick. Spread the caramelized onions to within 1/2-inch of the edge of the dough and top evenly with the cheeses, pine nuts and rosemary. Drizzle with extra-virgin olive oil and transfer to the pizza stone. Bake until the crust is brown around the edges, about 10 to 15 minutes.