Preheat and prep: Heat your oven to 425°F (220°C). Lightly grease a 9x13-inch baking dish.
Roast the cauliflower: Toss the florets with 1 tablespoon olive oil, a big pinch of salt, and pepper.
Spread on a sheet pan and roast 15–18 minutes, until just tender with a few browned edges. Set aside; lower the oven to 375°F (190°C).
Sauté the aromatics: While the cauliflower roasts, warm the remaining 1 tablespoon olive oil in a large skillet over medium heat. Add onion with a pinch of salt and cook 3–4 minutes until softened.
Stir in garlic for 30 seconds.
Cook the chicken: Add chicken to the skillet. Season with smoked paprika, Italian seasoning, onion powder, 1/2 teaspoon salt, and pepper. Cook 5–7 minutes, stirring, until the pieces are just cooked through.
Remove from heat.
Make the cheese sauce: In a medium saucepan, melt butter over medium heat. Whisk in flour and cook 1 minute. Slowly whisk in milk and chicken broth until smooth.
Simmer 3–4 minutes, stirring, until slightly thickened.
Enrich and season: Take the pan off heat. Whisk in Greek yogurt (or sour cream), 1 cup cheddar, mozzarella, and half the Parmesan until melted. Taste and adjust salt and pepper.
If you like heat, add the red pepper flakes.
Combine: Add roasted cauliflower and cooked chicken (with any juices) to the baking dish. If using spinach, peas, or roasted red peppers, fold them in now. Pour the cheese sauce over everything and gently toss or spread to coat evenly.
Top and bake: Sprinkle the remaining 1/2 cup cheddar and the rest of the Parmesan over the top.
Bake at 375°F (190°C) for 18–22 minutes, until bubbly and lightly golden.
Finish and serve: Let it rest 5–10 minutes so it sets. Garnish with chopped parsley or chives. Serve warm.