Heat oil in a large cast-iron or nonstick skillet over medium-high heat. Add onion, and sauté 5 minutes, stirring frequently. Stir in 1 teaspoon rosemary, salt, and pepper. Continue cooking 15 to 20 minutes or until onion is a deep golden brown, stirring frequently.
Preheat oven to 500°.
Roll dough into a 12-inch circle on a lightly floured surface. Place dough on a 12-inch pizza pan or baking sheet coated with cooking spray and sprinkled with cornmeal. Crimp edges of dough with fingers to form a rim. Top with onion. Bake at 500° for 10 minutes. Add cheese and walnuts; bake an additional 3 minutes or until cheese melts. Remove from oven; top with 1 teaspoon rosemary and arugula. Cut the pizza into 4 wedges.