Preheat oven to 350F (180C). Spray a 9- x 13-inch baking dish with cooking spray and set aside.
Melt 1 tablespoon of butter in a large skillet over medium-high heat. Add flour and stir to cook for about 1 minute. Whisk in broth, milk, garlic, and salt and bring to a simmer, stirring constantly. Allow sauce to cook until it thickens, whisking regularly so that the sauce is smooth (about 5-10 minutes).
Remove from heat, stir in cooked rice, chicken, broccoli, and 1 cup of the cheddar cheese.
Transfer chicken mixture to prepared dish.
Place cheddar snack crackers in a large zip-top bag. Seal the bag and use a mallet or rolling pin to crush the crackers into crumbs. You should have about ¾ cup of crumbs.
Melt the remaining 1 tablespoon of butter in a small bowl, add the cheddar cracker crumbs, and toss until the crumbs are coated.
Top casserole with remaining ½ cup cheddar cheese and sprinkle with buttered cracker crumbs.
Bake, uncovered, until bubbly and heated through (about 20-25 minutes).