Preheat the oven to 400°F (200°C) and spray a 9x13-inch baking dish with cooking spray.
Cut each large chicken breast into 3 equal pieces and set aside.
Place the Ritz crackers in a food processor and pulse until fine crumbs form; alternatively, crush in a sealed bag with a rolling pin.
Set up three shallow bowls: crumbs (seasoned with 1/4 tsp salt and 1/8 tsp pepper), milk, and grated cheddar cheese.
Dip each chicken piece first into the milk, then press into the grated cheddar so the cheese adheres, and finally roll in the seasoned cracker crumbs to coat.
Arrange the breaded chicken pieces in the prepared baking dish and sprinkle with dried parsley.
Cover the dish with foil and bake for 30–35 minutes, until an instant-read thermometer inserted into the thickest piece reads 165°F (74°C).
While the chicken bakes, whisk together the cream of chicken soup, unsalted butter, and sour cream in a medium saucepan over medium heat until warmed through and smooth.
When the chicken is done, spoon the warm cream sauce over the chicken and serve.