In a large saucepan, melt 2 tablespoons of butter over medium heat until bubbling. Add 1 diced onion and 2 minced garlic cloves, sauté for about 3-4 minutes until the onion is translucent and fragrant.

Stir in 1 cup of uncooked rice, ensuring it's well-coated with the buttery onion mixture. Toast the rice for an additional 2 minutes, stirring often to prevent sticking.

Pour in 2 cups of chicken broth and bring the mixture to a gentle boil. Once boiling, reduce the heat to low, cover the saucepan tightly, and let it simmer for 12-15 minutes.

After 15 minutes, add in 2 cups of broccoli florets, covering the pot again. Cook for another 5 minutes until the broccoli turns bright green and is tender yet still has a slight crunch.

Stir in 2 cups of cooked chicken, 1 cup of milk, 1 cup of shredded cheddar cheese, and 1/4 cup of grated Parmesan cheese. Stir continuously for about 3-5 minutes until the cheese melts completely and all ingredients are well combined.

Season with salt, pepper, and a pinch of cayenne if desired. Keep the stove on low as you heat everything through for an additional 2-3 minutes.

Remove from the stove and serve hot. You can add a sprinkle of extra cheese or fresh herbs on top for an appealing finish.