Preheat oven to 400° Fahrenheit. Heat medium saucepan over medium heat.
When saucepan is warm, add butter and let melt completely, then add chopped onion. Sauté onion until translucent and very soft, approximately 5 to 10 minutes.
Add garlic to saucepan. Sauté garlic, stirring occasionally, until fragrant, approximately 1 minute.
Sprinkle 6 tablespoons flour into saucepan with sautéed onions and garlic. Whisk well until all ingredients are combined, then cook mixture 5 minutes, stirring constantly, until mixture is strong yellow color.
Pour in chicken stock and whisk until ingredients are thoroughly combined. Increase heat to medium-high and bring mixture to low boil. Once mixture begins to boil, reduce heat back to medium and let mixture simmer 5 to 10 minutes, or until mixture has thickened enough to coat back of spoon.
When mixture has thickened, add dried thyme, salt, and pepper. Whisk well to incorporate spices.
Reduce heat under saucepan to low. Add 2 cups shredded cheese and whisk until cheese is melted and well incorporated. Remove saucepan from heat and set aside.
Add chicken, rice, and broccoli to 9x13 casserole dish, then pour in cheese sauce mixture. Stir to combine ingredients and spread mixture out into one even layer.
Sprinkle remaining 1 to 2 cups shredded cheese evenly over top of chicken and rice mixture.
Place casserole dish in preheated oven. Bake, uncovered, 20 minutes, or until broccoli can be easily pierced with tines of fork.
When broccoli is fork-tender, remove casserole dish from oven and let stand 5 minutes. After 5 minutes, portion casserole onto serving plates or bowls and serve warm.