Prep the oven and pan: Heat oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Blanch or steam the broccoli: Bring a pot of salted water to a boil.
Add broccoli and cook 2–3 minutes until bright green and just tender. Drain well and pat dry. If using frozen, thaw and squeeze out excess moisture.
Sauté the garlic: In a skillet, warm olive oil or butter over medium heat.
Add garlic and cook 30–60 seconds until fragrant. Don’t let it brown.
Make the sauce base: Reduce heat to low. Whisk in cream cheese until smooth and melty.
Add Greek yogurt, almond milk, Dijon, onion powder, thyme, and smoked paprika. Stir until creamy. Season with salt and pepper.
Add the cheeses: Stir in 1/2 cup mozzarella and all the Parmesan until just melted.
Taste and adjust seasoning.
Combine: In a large bowl, mix shredded chicken and broccoli. Pour the sauce over and toss to coat evenly.
Assemble the bake: Transfer mixture to the baking dish. Top with remaining 1/2 cup mozzarella.
Add a pinch of red pepper flakes if you like heat.
Bake: Cook for 18–22 minutes, until bubbly around the edges.
Broil for color: Switch to broil for 1–3 minutes to brown the top lightly. Watch closely to avoid burning.
Finish and serve: Rest 5 minutes. Sprinkle with chopped parsley or chives.
Serve warm.