Heat a wood-fire oven to 700 degrees F or a regular oven to 400 degrees F.
For the pesto: Pulse the basil in a food processor, and then add the oil, followed by the nuts, garlic and salt, pulsing continuously and adding subsequent ingredients slowly until almost smooth. Remove from the processor and stir in the Parmesan.
For the pizza: Spread the pesto onto the pizza crust. Follow with a layer of the mozzarella. Top with the goat cheese and tomatoes. Bake until the crust begins to turn golden and the cheese has melted.