Preheat oven to 375 degrees F (190 degrees C). Grease a baking sheet.
Measure 2 tablespoons of the beaten egg into a small bowl and set aside. Mix remaining egg, tuna, onion, 1/2 cup Cheddar cheese, parsley, celery salt, 1/4 teaspoon black pepper together in a large bowl.
Stir 2 cups baking mix and cold water together to form a soft dough. Cover a cutting board with cheesecloth and lightly dust with flour. Turn dough out onto the prepared cutting board and knead 5 times. Roll into a 15×10-inch rectangle. Spread tuna mixture over the dough. Roll, beginning at one of the long side, into a cylinder; transfer to the prepared baking sheet with the sealed edge down. Shape the cylinder into a ring and pinch the ends together. With scissors, make cuts 2/3 of the way through ring at 1-inch intervals. Brush top with reserved egg.
Bake in preheated oven until the outside is golden brown, 25 to 30 minutes.
Melt butter in a saucepan over low heat. Beat 1/4 cup baking mix, salt, and 1/4 teaspoon pepper into the butter; cook until smooth and bubbly, about 5 minutes. Remove saucepan from heat and stir milk into the butter mixture until smoothly incorporated. Bring the mixture to a boil, stirring continually; cook at a boil for 1 minute; remove from heat and stir 1 cup Cheddar cheese into the mixture until melted. Pour over the tuna ring casserole to serve.