Make the vegan Parmesan cheese: In a food processor, combine the cashews, nutritional yeast, garlic powder, salt, and pepper. Pulse until the texture is crumbly and resembles Parmesan cheese. Store in an airtight container for up to 1 month.
Make the pesto: In a food processor, combine the basil, arugula, walnuts, garlic, lemon juice, salt, pepper, and olive oil. Process until smooth. Store in an airtight container for up to 1 week.
Preheat the oven to 425°F (220°C).
Use your hands to stretch the pizza dough into a circle about 14 inches (35 cm) in diameter and ½-inch (1 ¼ cm) thick. Lay on a sheet of parchment paper.
Spread half of the pesto over the pizza dough in an even layer. Arrange the tomato slices on top. Transfer the pizza on the parchment to a baking sheet.
Bake for 14-16 minutes, until the edges of the pizza are lightly golden. Let cool for 5 minutes.
Drizzle more pesto over the pizza and and sprinkle with vegan Parmesan. Cut into 8 slices and serve.