Preheat your oven to 350°F (175°C). Grease a 9x13-inch baking dish with cooking spray or olive oil.
In a pot of boiling salted water, blanch the broccoli florets for about 3 minutes until bright green and slightly tender. Drain and rinse with cold water to stop the cooking process. Set aside.
In a skillet over medium heat, heat the olive oil. Add the chopped onion and garlic, cooking until the onion is translucent (about 3-5 minutes).
In a large mixing bowl, combine the cooked quinoa, sautéed onion and garlic, cream of mushroom soup, vegetable broth, thyme, garlic powder, paprika, salt, and black pepper. Stir in half of the cheddar and mozzarella cheese, and the blanched broccoli florets. Mix until well combined.
Add the stuffing mix to the mixture and stir until the stuffing is evenly distributed and soaked in the liquid. Pour the mixture into the greased baking dish and spread evenly.
In a small bowl, mix the milk with the remaining cheese. Pour this mixture over the casserole, ensuring an even distribution.
Cover the dish with aluminum foil and bake in the preheated oven for 25 minutes. After 25 minutes, remove the foil and bake for an additional 15-20 minutes, or until the top is golden and bubbly.
Allow the casserole to cool for a few minutes before serving.