For the pesto: Add the walnuts, olive oil, sea salt, garlic, basil leaves and lemon juice to a food processor and pulse until the mixture forms a paste.
For the pizza: Preheat the oven to 500 degrees F.
Dust a 16-inch pizza pan or a rimmed baking sheet with cornmeal and press the pizza dough into the pan. Leave to rest.
Coat a large saute pan with the olive oil and set over high heat. Add the mushrooms, Brussels sprouts, garlic, red bell pepper, orange bell pepper and onion and saute until the veggies begin to soften, about 3 minutes. Lower to medium heat, add the sun-dried tomatoes and capers and saute for another 2 minutes.
Spread the dough evenly with 1/4 cup of the pesto, leaving a 1/2-inch border around the edges. Distribute the sauteed veggies evenly over the pesto. Scatter the mozzarella over the veggies.
Bake until the crust is nicely browned and the cheese has melted, 15 to 20 minutes. Garnish with basil leaves, then cut into wedges. Serve with red pepper flakes, garlic powder and dried oregano.
If you prefer your pizza extra crispy, preheat the pizza pan in the oven. Assemble the pizza on a cornmeal-dusted piece of parchment paper and transfer the pizza and parchment to the preheated pizza pan.