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Preheat the oven to 425°F/ 200°C. Line a round 14" pan with parchment paper; set aside.
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Place the broccoli florets in a bowl of a food processor. Pulse until the broccoli is grated and looks like rice or couscous, being careful not to over-pulse. Place the broccoli in a steaming basket and steam the broccoli for 2 minutes, or until softened. Place the steamed broccoli on an open cheesecloth or dish towel and allow it to cool. When it’s cool to the touch, wrap the broccoli in the towel and squeeze out all the excess water, so it is as dry as possible.
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While the broccoli is cooling down, prepare the sauce. Heat olive oil in a small saucepan over medium heat. Whisk in flour and continue whisking for 1-2 minutes (do not let it brown). Slowly add milk, whisking constantly until smooth, thickened, and bubbly. Season to taste and add garlic. Add Kefalotyri/ Parmesan cheese and whisk for another 30 seconds. Remove from heat and continue whisking just until cheese is melted and sauce is smooth, then set aside to cool before using for pizza.
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Transfer the broccoli to a large bowl, add the eggs, 1 cup of mozzarella cheese, Kefalotyri/ Parmesan cheese, oregano, garlic, salt, and pepper. Mix with a spoon or your hands until all the ingredients are well blended and a “dough” forms. Place the dough on the prepared baking sheet, and shape it into a round pizza crust, about ½ inch thick.
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Bake the dough in the preheated oven for 10-15 minutes until golden. Remove from the oven and add the pizza sauce, remaining mozzarella cheese, and mushrooms. Return to the oven and bake until the cheese is melted and the crust is crisp, about 10 more minutes. Cool for a few minutes before cutting. Add the avocado slices, sprinkle with parsley and serve.