Preheat oven to 400°F and spray a 9x13-inch baking dish with nonstick spray.
Place crushed Ritz crackers into a shallow dish. In another dish, add the milk. In a third dish, place the shredded cheddar cheese.
Dip each chicken breast into the milk, then the cheese, and then coat with cracker crumbs. Press crumbs firmly to adhere.
Place coated chicken breasts into the prepared baking dish. Sprinkle parsley over the top.
Cover with foil and bake for 35 minutes. Remove foil and bake an additional 10–15 minutes until golden and crispy.
To make the sauce: In a saucepan, melt butter over medium heat. Whisk in flour and salt, stirring until bubbly.
Gradually whisk in chicken broth and milk. Cook and stir until thickened, about 5 minutes.
Serve sauce over crispy cheddar chicken.