If the dough has been refrigerated, transfer it to the countertop to let it rise for about 1 hour before making pizza.
Preheat the broiler with the rack set 5 inches from the element or flame. If you are using a cast-iron skillet or griddle pan for the pizza, set it over medium-high heat until it gets smoking hot, about 15 minutes. Transfer the skillet (turned upside down) or griddle pan to the broiler. If you are using a baking stone, heat it in a 500°F oven for an hour, then carefully transfer it to the broiler.
Put the figs in a small skillet set over medium heat, pour in the wine, and bring to a boil. Turn off the heat and let the figs soak for at least 30 minutes. Drain, then chop into 1/2-inch pieces.
Toast the walnut pieces in a small, dry skillet over medium-high heat, shaking the skillet frequently, until they are very fragrant and starting to brown, 3 to 4 minutes. Immediately transfer to a plate, let cool, and then coarsely chop.
To shape the dough, dust a work surface liberally with flour and put the ball of dough on it. Sprinkle with flour and knead a few times until the dough comes together and holds its shape when you form it into a ball. Add more flour if necessary. Form it into an 8-inch round by pressing from the center out toward the edges, leaving a 1-inch border thicker than the rest.
Make sure you have all the topping ingredients measured out and ready before you assemble the pizza, because once you place the dough on the cooking surface you can't easily move it.
Open the oven or broiler door, and quickly slide out the rack with the cooking surface (skillet, griddle pan, or baking stone) on it. Pick up the dough and quickly transfer it to the cooking surface, pressing it back into shape if need be, while being careful not to touch the cooking surface with your fingers.
Drizzle 1 Tbs. of the oil onto the dough, scatter the walnut pieces on top, then the radicchio, then the chopped figs, and then the cheese. Slide the broiler rack back into the oven and close the door.
Broil the pizza until the crust has puffed up around the edges, the pizza has blackened in spots, and the cheese has melted, 3 to 4 minutes.
Remove the pizza with a wooden or metal pizza peel or a square of cardboard, transfer it to a cutting board, and let it rest for a few minutes. Drizzle the remaining 1 Tbs. of oil on top, cut the pizza into quarters, transfer it to a plate, and eat.