Make pizza dough: In a large bowl stir together the water and yeast. Stir in 1 cup of flour and 2 tablespoons avocado oil, until smooth, set aside for 5 minutes until little bubble starts to rise to the top.
In another bowl, combine 2 cups flour, sugar, and salt. Stir the dry ingredients into the yeast bowl until a soft ball of dough forms. Transfer dough to a lightly floured surface, and knead dough 60 times or until dough is smooth. Shape into a ball, place in a bowl that has 1 tablespoon of oil, place dough in top side down to coat with oil, flip over, cover with a towel or plastic wrap, and place in a warm spot for 40 minutes to 1 hour, until the dough has doubled in size.
While the dough is rising, make pesto. In a food processor, add the basil, lemon zest, lemon juice, garlic, 1 cup parmesan, kosher salt, pepper, and chili flakes. Pulse 10 times, add the olive oil and blitz on high for 30 seconds, or until pesto is smooth.
Make mushrooms: Turn on the heat to medium in a medium pot, add the oil, wild mushrooms, salt, and black pepper, cook for 8 minutes, stirring occasionally. Place on a plate lined with paper towels to drain.
Second proof: Pour 4 tablespoons of avocado oil on a baking sheet and spread evenly over the baking sheet pan and rim. Lightly dust a surface with flour, punch dough down and flatten into a disc shape. Then with one hand slightly turn the dough, while using the other hand to form dough into a disc shape. Do this about 12 times, pick the dough up and let gravity do 99% of the work and turn to let dough stretch. Place dough on the baking sheet and mold the dough to fit the baking sheet. If the dough tears pinch back together. Cover and place in a warm spot for 40 minutes to 1 hour, untildough has double in size.
Preheat grill to 500 degrees. After the dough has finished rising the second time, use your fingers to release the air from the dough, and create a 1-inch border to be your crust. Evenly spread ½ cup parmesan, then spread pesto over the dough, leaving a 1-inch border from edge. Then add mozzarella, mushrooms, and ¼ cup of parmesan. Bake in a closed grill for 10 to 12 minutes.
Preheat heat broiler to hi. Place oven rack 1 level away from the broiler. When the pizza is finished on the grill place under the broiler for 1 minute or until cheese is bubbling brown. Transfer pizza to cutting board, cut pizza, garnish with remaining parmesan, and basil. Serve hot.