Melt butter in a small saucepan over medium heat. Whisk in flour until a thick, creamy paste forms. Whisk in milk mixture until sauce is thick and smooth, about 5 minutes. Remove from heat and let cool.
Whisk eggs in a large bowl. Add chicken; mix until coated with eggs. Mix in 1 cup corn bread stuffing and parsley. Stir in sauce until thoroughly combined. Cover bowl with plastic wrap.
Freeze chicken mixture until firm enough to form into balls, about 30 minutes.
Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish with cooking spray.
Mix remaining 1 cup corn bread stuffing and Parmesan cheese together in a bowl.
Scoop up a tablespoonful of chicken mixture; roll in corn bread-Parmesan mixture and form into a ball. Place on the greased baking dish. Repeat with chicken mixture.
Bake in the preheated oven until golden and crisp, about 30 minutes.
Optional variations: I've added 1 tablespoon honey to the butter-flour mixture to slightly sweeten it. I've drizzled melted butter over the cooked croquettes as well. Margarine can be used instead of butter but I've found that the mixture doesn't get firm enough to make balls.