In a large skillet, melt 2 tablespoons of butter over medium heat. Add the chopped onion and cook for 2 minutes. Shift the onions to one side of the skillet, and add the chicken pieces. Cook until the chicken is browned on both sides, flipping halfway through. Remove the browned chicken and set it aside on a plate. Stir the onions as needed. Once all the chicken is cooked, add the minced garlic and sauté for 1 minute, stirring constantly. Transfer the onions and garlic to the plate with the chicken.
In a separate saucepan, combine 2 cups of water, 1 tablespoon of butter, and ½ teaspoon of salt. Bring to a boil, then stir in the uncooked rice. Cover and reduce heat to low, simmering for 12-15 minutes until the water is absorbed. Meanwhile, steam the broccoli either on the stovetop or in the microwave until tender.
Preheat your oven to 375°F (190°C). Lightly grease a 9x13-inch casserole dish with nonstick cooking spray.
In the same skillet, melt 4 tablespoons of butter over medium-low heat. Sprinkle in the flour, whisking continuously for 2-3 minutes. Add the crushed red pepper, onion powder, dried marjoram, and dried thyme. Gradually whisk in the chicken broth and milk, alternating between the two until the mixture is smooth and slightly thickened. Reduce the heat to low and stir in 1 ¼ cups of shredded cheddar cheese until melted and creamy.
Stir the cooked chicken, rice, and steamed broccoli into the cheese sauce, ensuring all ingredients are evenly coated.
Pour half of the chicken mixture into the prepared casserole dish. Sprinkle ½ cup of cheddar cheese on top, then add the remaining chicken mixture and top with the rest of the cheese. Combine the crushed Ritz crackers with 4 tablespoons of melted butter, and sprinkle this mixture evenly over the casserole.
Bake in the preheated oven for 20-25 minutes, or until the topping is golden brown and the cheese is fully melted.