Prep your oven and pan: Preheat oven to 375°F (190°C).
Grease a 9x13-inch casserole dish with a little oil or cooking spray.
Sauté aromatics: In a large skillet over medium heat, add olive oil. Cook onion with a pinch of salt for 3–4 minutes until soft. Add garlic and cook 30 seconds until fragrant.
Cook the veggies: Add riced cauliflower and broccoli to the skillet.
Season with salt, pepper, smoked paprika, and thyme. Cook 5–7 minutes, stirring, until the cauliflower loses excess moisture and the broccoli turns bright green and slightly tender.
Make the creamy base: In a large bowl, whisk together Greek yogurt, softened cream cheese, almond milk (or broth), Dijon, and eggs until smooth. Stir in half the cheddar and half the mozzarella.
Add red pepper flakes if using. Taste and adjust salt and pepper.
Combine: Stir the shredded rotisserie chicken into the creamy base. Fold in the cooked veggie mixture until everything is evenly coated.
Assemble the casserole: Spread the mixture into the prepared dish.
Top with the remaining cheddar and mozzarella.
Bake: Bake uncovered for 22–28 minutes, until the edges are bubbly and the top is lightly golden. If you want deeper color, broil for 1–2 minutes at the end, watching closely.
Rest and garnish: Let the casserole rest 8–10 minutes so it sets. Sprinkle with chopped parsley or chives.
Slice and serve warm.