Preheat oven to 350 degrees F. Line a large baking sheet with foil or parchment paper. Place pizza crust on the prepared baking sheet.
Place cashews in a food processor and pulse until coarsely chopped. Add chickpeas, garlic, 1 Tbsp. oil, salt, and pepper; pulse to combine. Add 1/2 cup basil and pulse to a sandy consistency, 2 to 3 times.
Brush the remaining 1 Tbsp. oil over the pizza crust. Spread the chickpea mixture evenly over the crust. Top with spinach, tomatoes, mozzarella, and goat cheese.
Bake until the cheese is melted and the crust is crisp, 15 to 20 minutes. Garnish with the remaining 1/2 cup basil. Cut the pizza into 8 slices to serve.
To make ahead: Prepare chickpea base (Step 2) and refrigerate for up to 1 day.