Preheat a pizza oven or oven to 450 degrees F and place a pizza stone on the middle shelf.
Combine the goat cheese, mascarpone, ricotta, basil, oregano, garlic, lemon zest, salt and pepper in a food processor and blend until smooth.
Divide the dough into 2 pieces and roll out on a lightly-floured surface. Place 1 piece of dough on a pizza peel and drizzle the dough with 2 tablespoons olive oil. Smear half the cheese on evenly. Freshly grate about 1/4 cup Parmigiano-Reggiano on top. Bake until the crust is crispy, about 10 minutes. Repeat with the remaining dough.
Fan out the avocado on top of the cheese. Drizzle with olive oil, then sprinkle with sea salt and crushed red pepper, to taste.