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Preheat oven to 400 degrees and line two pizza pans with parchment paper.
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In a large food processor, process the cauliflower until a fine crumbled texture is reached. Do this in several batches if necessary.
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Place the cauliflower into a large bowl and microwave for about 10 minutes, stirring halfway through. Let stand until cool enough to handle, about 15 minutes.
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Place cauliflower into a clean kitchen towel or cheesecloth and wring out as much excess moisture as possible. This task also may need to be done in batches.
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Transfer the cauliflower back into a large bowl and add in the garlic, salt, pepper, oregano, crushed red pepper and 1 1/3 cup Parmesan. Stir until well combined and then add the egg whites, mixing until well combined.
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Divide the cauliflower in half and spread onto on the pizza pans, forming a ridge along the outside for the crust.
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Bake until golden brown, about 30 minutes.
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While pizza crust is cooking, assemble the toppings. In a small skillet over medium heat, add butter and apples. Cook for 3-4 minutes, stirring, then add brown sugar and continue to cook for another 2-3 minutes or until apple is browned and caramelized.
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Once the pizza crusts are cooked, remove from the oven and preheat the broiler to high heat. Divide remaining 1/4 cup of Parmesan over each pizza crust (2 tablespoons each). Top each pizza with 1/2 cup smoked mozzarella, 2 tablespoons red onion and half of caramelized apples.
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Broil the pizzas just until the cheese melts, about 2-3 minutes. Remove from oven and top with fresh arugula and a drizzle of olive oil just before serving.