Place a baking sheet in the oven. Preheat oven to 475° (keep baking sheet in oven as it preheats).
Brush crust with 1 tablespoon olive oil; top pizza evenly with shredded chicken and ricotta cheese. Sprinkle with goat cheese, black pepper, and red pepper. Carefully place pizza on preheated baking sheet; bake at 475° for 10 minutes.
Combine white wine vinegar, remaining 1 tablespoon olive oil, and Dijon mustard in a medium bowl, stirring with a whisk. Add arugula and radish slices; toss to coat. Top pizza with arugula mixture. Cut the pizza into 8 wedges.