Arrange four shallow dishes or pie pans as follows: Meat (season lightly with salt and pepper if desired), 2 cups flour stirred together with 2 teaspoons salt and 1 teaspoon pepper, milk-egg mixture, beaten together, and finally a clean plate or tray for the coated meat.
Start with the meat in the egg mixture, turn to coat both sides. Put it into the seasoned flour, turn over to cover evenly. Repeat coating with egg then flour. Place on wax paper (or parchment) and let dry for ½ hour.
Heat oil in a large skillet over medium-high heat. Add the meat pieces without crowding. Cook until edges are golden (2-3 minutes). Turn over and cook for an additional 2-3 minutes. Remove to an oven proof plate and keep warm. Repeat with additional pieces.
Remove all but about ¼ cup oil from the pan. Reduce heat to medium-low. Sprinkle about 1/3 cup of your seasoned flour evenly into the pan and whisk until mixture is golden brown. If the mixture seems oily add additional flour a tablespoon at a time. Keep whisking and add 2 cups of milk. Let the gravy come to a slow boil. It will thicken gradually, about 5-10 minutes. Season to taste with additional salt and pepper, if desired.
Serve with mashed potatoes and spoon the gravy over both.