Making the croquettes: In a heavy pot, heat the milk and stock on the stove and add the white pepper, parsley, and salt. In a separate pot, melt the butter and add the celery. Stir in the flour and cook for 3 minutes. Pour in the hot milk/stock mixture and stir until thickened and smooth. Fold in the chicken and allow to cool.
Breading the croquettes: Shape 3 oz portions of the croquette batter into small balls and place on a tray lined with wax paper. Set out three bowls/breading dishes. In the first, make an egg wash with the egg, milk, and salt. The second bowl is flour, the third is breadcrumbs. Dip the croquettes into the egg wash, then flour, then egg wash again, then breadcrumbs.
Cooking the croquettes: Fry the croquettes in about 1 inch of oil, heated to 350 degrees until golden brown, about 2-3 minutes per side. Drain well and blot on paper towels.
Cooking the gravy: Melt the butter in a 1 or 2 qt saucepan. Stir in the flour. Pour in the stock and whisk until smooth. Bring to a boil, reduce heat, and cook for 20 minutes, whisking occasionally. Add the parsley, white pepper, and salt to taste.
Serve over mashed potatoes, noodles, or rice.