Heat the oven to 375°F and grease a 9-by-13-inch baking dish with 1 teaspoon butter.
Whisk the condensed soup, milk, mustard, garlic powder and salt in a large bowl.
Fold in the rice, cooked chicken, broccoli and 1 1/2 cups cheddar and spread the mixture in the dish.
Combine the crackers, melted butter and remaining cheddar and scatter the crumbs evenly on top.
Bake uncovered until bubbling around the edges and browned on top, 25 to 30 minutes.
Rest the casserole for 8 minutes before scooping so the filling holds together.