Preheat oven to 350°F. Spray a 3-quart or 9 x 13-inch baking dish with cooking spray and set aside. Melt 1 tablespoon butter in a large skillet over medium-high heat. Add the onion, bell pepper, and celery. Cook, stirring frequently, until the vegetables are soft, about 7-8 minutes. Add the garlic; cook for 1 more minute, stirring constantly. Remove from the heat.
In a large bowl, stir together rice, shrimp, condensed soup, chicken broth, and mayonnaise. Add the cooked vegetables; stir gently to combine. Taste and season with salt and pepper, if necessary.
Transfer the shrimp and rice mixture to the prepared baking dish. Sprinkle cheese on top.
In a separate bowl, stir together Ritz cracker crumbs and melted butter. Sprinkle over top of the casserole.
Bake, uncovered, for about 25 minutes (or until hot, bubbly, and golden brown on top).