Add the oil to a large heavy bottomed skillet, over medium heat.
Meanwhile, prepare the beef by pounding each piece to tenderize. Put 1 1/4 cups of the flour in a shallow dish and season well with salt and pepper, to taste. Add the egg mixture to a shallow dish and season with salt and pepper, to taste. Dredge the steaks in seasoned flour, shaking off any excess, then dip them in the egg mixture and back into the flour.
Once oil is hot, add the steaks and fry until golden, about 3 to 4 minutes per side. Be sure to not overcrowd the pan and do in batches, if necessary. Remove the fried steaks to a paper towel lined dish and set aside. Sprinkle the remaining 1/4 cup of flour into the cooking oil and whisk until fully incorporated. Cook over medium-high heat until just slightly darker in color, about 3 minutes. While whisking, pour the milk in a slow steady stream. Bring to a boil, then reduce the heat and simmer until thick, another 5 to 6 minutes. Taste the gravy for seasoning and adjust as necessary. Arrange the steaks on serving plates with the gravy spooned over the top. Enjoy immediately.