Melt the butter in a large, high-sided skillet over medium heat. Add the celery and onion and sauté for 5 minutes, until soft.
Sprinkle the flour over the vegetables and stir for 1 minute, then slowly add the milk, stirring constantly to prevent clumps from forming in the sauce. Cook for about 5 minutes, stirring occasionally, until the sauce thickens.
Stir in the chipped beef and a tablespoon or so of chopped parsley, and season with a few grinds of black pepper.
Serve immediately over cooked noodles, Tater Tots, hash browns, baked potatoes, or toast.