Heat the oven: Preheat to 375°F (190°C). Lightly coat a 9x13-inch baking dish with cooking spray.
Cook the pasta: Boil in salted water until just shy of al dente.
Drain and set aside. It will finish in the oven.
Sauté aromatics: Warm olive oil in a skillet over medium heat. Cook onion with a pinch of salt until soft, 4–5 minutes.
Stir in garlic for 30 seconds until fragrant. Remove from heat.
Make the sauce: In a large bowl, whisk Greek yogurt, cottage cheese, and cream cheese until mostly smooth. Stir in chicken broth, salt, pepper, Italian seasoning, and smoked paprika.
Add the cheese: Fold in mozzarella, cheddar, and Parmesan, reserving a small handful of mozzarella and Parmesan for topping.
Combine the base: Add cooked chicken, sautéed onion/garlic, broccoli, and pasta to the bowl.
Gently mix until everything is evenly coated.
Assemble: Spread the mixture into the prepared baking dish. Sprinkle the reserved cheeses on top, then scatter panko for crunch. Lightly mist the topping with cooking spray.
Bake: Bake uncovered for 20–25 minutes, until bubbling at the edges and the top is golden.
Rest and serve: Let it sit for 5–10 minutes so it sets and slices cleanly.
Garnish with chopped parsley or chives if you like.