Preheat the oven to 375 F.

Add the butter to a saute pan over medium heat, once melted, add the onion and garlic and cook for 3-4 minutes, until softened.

Add the flour to the pot and whisk to combine, then slowly pour in the chicken broth, whisking the mixture constantly to ensure there are no lumps. Once smooth, whisk in the milk, salt, pepper, and thyme and let cook for 1-2 minutes, until the sauce is thick and bubbling.

Add the rice, chicken, and broccoli to a 4-quart baking dish, then pour the sauce over and toss to combine. Make sure that the sauce completely covers the rice mixture, if it doesn't, add another ½ cup of broth or switch to a different dish. Cover the dish with foil and make sure that the foil is secured very tightly. It needs to steam to ensure the rice cooks through.

Bake for 50 minutes, until the rice is fully cooked through and the liquid is absorbed, then remove the foil and sprinkle with cheese and bake for an additional 10-15 minutes, until cheese is melted.