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Heat oil in a cast iron skillet to 360 degrees.
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Cut the steak into 4 pieces. Pound out the meat.
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Season with salt and pepper.
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Combine the egg and 3 Tbsp of milk. Put 1 1/2 cups of the flour in a pan and season with salt and pepper.
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Dredge the steaks in the flour, coating each piece evenly and tapping off excess. Dip the steak in the egg wash and dredge again in the flour.
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Fry steaks in hot oil until golden brown on each side, about 3 minutes. Remove and drain on paper towels.
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Season with salt and pepper.
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Pour off all but 1/4 cup of the oil.
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Over medium heat add the 3 Tbsp of flour and cook for 3 to 4 minutes, whisking constantly.
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Add the remaining 3 cups of milk, 1/2 cup at a time, whisking constantly.
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Bring to a boil then reduce the heat to medium low. Season with salt and plenty of pepper.
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Cook 8 to 10 minutes, whisking constantly. The gravy should be thick enough to coat the back of a spoon.