Divide the meat into 4 pieces, and flatten into thin beef patties. Heat a heavy skillet over medium heat until almost smoking, and cook the beef patties until browned on both sides, about 4 minutes per side; set the cooked burgers aside. Drain excess grease but leave any browned drippings in the skillet.

In a shallow bowl, mix together 1 cup of flour with salt and black pepper. In a separate shallow bowl, beat the eggs until smooth, mixing in a half teaspoon of water if needed.

Heat the vegetable oil in the skillet over medium heat. Dip both sides of cooked burgers into the eggs, then dredge in the seasoned flour; repeat 3 times to build up a thick coating on the burgers. Place the burgers into the hot oil, and pan-fry until golden brown on both sides, about 5 minutes per side. Set burgers aside.

Spoon away any excess vegetable oil. Melt butter in the skillet over medium heat, and stir to loosen any brown flavor bits left in the bottom of the skillet. Whisk in 1 tablespoon of flour until smooth. If you prefer a very thick gravy, use more flour. Gradually whisk in milk, cooking until the gravy is thick. Season to taste with salt and black pepper, and serve the gravy over the chicken-fried beef patties.