Wide ribbons of fresh butternut squash become tender while the pizza bakes, a colorful and faintly sweet contrast to the peppery arugula pesto.
Preheat oven to 450°F.
Place crust on a baking sheet; coat with cooking spray. Bake at 450°F for 5 minutes.
Process arugula, parsley, walnuts, juice, garlic, and Parmesan cheese in a mini food processor until finely chopped. With food processor running, slowly add 2 1/2 tablespoons oil. Add 1 tablespoon water; pulse to combine. Spread mixture over crust, leaving a 1/2-inch border.
Combine remaining 1 1/2 teaspoons oil, squash, salt, and pepper in a bowl; arrange over pesto. Sprinkle goat cheese over pizza. Bake at 450°F for 10 minutes. Cut into 8 slices.