Place cauliflower florets in a blender and cover with water; blend briefly on high speed until cauliflower is the size of rice.
Pour cauliflower rice and water into a pot; add 1 tablespoon salt. Bring to a boil and cook for 1 minute. Drain cauliflower rice in a wire sieve or colander lined with cheesecloth; cool.
Heat oil and butter in a skillet over medium heat; stir in onion and celery. Cook and stir until the onion has softened and turned translucent, about 5 minutes.
Beat eggs, half-and-half, remaining 1 tablespoon salt, black pepper, onion powder, and garlic powder in a large bowl. Stir in cauliflower rice, onion mixture, chicken, broccoli, mozzarella cheese, and pepper Jack cheese. Spread into a 9x13-inch baking dish. Sprinkle top with Parmesan cheese.
Bake in the preheated oven until the sauce is bubbly and the top is golden brown, about 50 minutes.