Heat the oil in a heavy 9-inch cast iron skillet, to 360 degrees F.
Using a meat mallet, pound out the meat. Season the steak with salt and pepper.
Combine the egg with 3 tablespoons of the milk. Put 1 1/2 cups of the flour in a shallow pan and season with salt and pepper. Dredge the steaks in the flour, coating each piece evenly and tapping off any excess. Drip the steak in the egg wash, coating it completely and letting the excess drip off. Dredge again in the flour, shaking off any excess.
Fry the steaks in the hot oil, until golden brown on each side, about 3 minutes. Remove and drain on paper towels. Season with salt and pepper.
Carefully pour off the oil, leaving behind about 1/4 cup of the oil along with the brown bits. Over medium heat, add the remaining 3 tablespoons flour and cook for 3 to 4 minutes, whisking constantly.
Add the remaining 3 cups milk, 1/2 cup at a time, whisking constantly. Bring to a boil, then reduce the heat to medium-low.
Season with salt and plenty of pepper. Cook for 8 to 10 minutes, whisking constantly. The gravy should be thick enough to coat the back of a spoon. If it is too thick, add a little water to thin it.