Heat the oil in a skillet over a medium heat. Add the mushrooms and onions and cook until tender and most of the liquid evaporates, about 5 minutes. Stir in the garlic and remove from the heat.
Place the pizza crust on a baking sheet. Spread the sauce over the crust leaving a 1-inch border. Top with the mushroom mixture and sun-dried tomatoes. Sprinkle with the mozzarella and Parmesan cheeses. Bake until the crust is crisp and the cheese is nicely melted, about 13 minutes. Sprinkle with basil, cut into 8 slices and serve.