Preheat oven to 400º F. In a medium saucepan over medium heat, melt butter. Add chopped onion and sauté until translucent and very soft, almost browning, about 5-10 minutes. Add garlic and sauté, stirring constantly, for 1 minute, or until very aromatic.

Sprinkle flour over and whisk to combine. Cook about 5 minutes, stirring constantly, until flour is cooked and mixture is a strong yellow color. Add stock and whisk to combine. Bring to a boil, then reduce heat to a strong simmer. Cook for 5-10 minutes, or until mixture is thickened and coats the back of a spoon. Stir in thyme leaves and salt and pepper, to taste. Reduce heat and stir in 2 cups cheese. Whisk until very well combined and remove from heat.

In an ungreased 13x9" pan, combine chicken, brown rice, and broccoli then pour sauce over. Stir until combined. Sprinkle remaining 1-2 cups cheese over top. Bake uncovered for 20 minutes, or until a fork easily pierces a piece of broccoli. Remove from oven and let stand 5 minutes before serving.