Step 1Place a cast-iron pizza pan on the middle rack of the oven and heat to 450 degrees. Scatter the walnuts over the hot pan and toast until browned and fragrant, about 5 minutes, stirring frequently to make sure they don't burn. Carefully take the hot pan from the oven, remove the toasted nuts, and return the pan to the oven. Break apart the walnuts if they're large.
Step 2Brush the prepared pizza dough with 1 teaspoon walnut oil. Spread the arugula over the dough, then crumble the goat cheese over the top. Sprinkle the walnuts over the pizza, drizzle with the remaining walnut oil and grind black pepper over the top.
Step 3Slide the parchment paper with the pizza onto an inverted cookie sheet or peel, then transfer both parchment and pizza onto the hot pan in the oven. Cook until golden and crisp, 12 to 16 minutes, rotating the pizza once during baking. Remove the pizza from the oven by pulling out the parchment paper and sliding it back onto the cookie sheet, or remove the hot pizza pan and place on a trivet. Drizzle with a little balsamic vinegar, slice and serve immediately.