Preheat the oven to 400°F.
Using a food processor or powerful blender, process the first 6 ingredients (basil through salt) until well mixed. Add the goat cheese and process until the goat cheese it thoroughly incorporated. Set aside.
Slice the tomatoes to about 1/4-inch thickness. Place them single layer onto a double thickness of paper towels. Sprinkle with 1/4 teaspoon salt and cover with another double thickness of paper towels.
Line a baking sheet with parchment paper, and place the pizza crust on top. Spread 3/4 cup of the goat cheese and walnut pesto onto the pizza crust, leaving about 1/2-inch margin along the edge. Refrigerate the remaining 1/4 cup pesto for another use.
Bake for 5 minutes.
Gently pat the top of the sliced tomatoes to remove excess moisture. Place the tomatoes over the pesto. Sprinkle with black pepper. Bake for 8 minutes or until the tomatoes are soft and the crust is cooked through.
Remove from the oven. Sprinkle with chopped basil and fresh arugula. Slice and serve.