Cut the round steak into 4 pieces. With the textured side of a meat tenderizer (OXO Good Grips Meat Tenderizer), pound each piece (on a meat-only dishwasher-safe cutting board) until almost double in size and about 1/4 inch thick.
In shallow dish combine the flour with 1/4 teaspoon each of salt, coarsely ground black pepper and cayenne. In another shallow dish beat together the egg and milk.
Working with one piece at a time, season both sides with salt and pepper, dip into the egg mixture, dip into the flour mixture, dip into the egg mixture again, dip into the flour mixture again. Really press the flour in. This really goops up the hands, so just use one hand and keep one hand clean; you'll need it.
When all meat is coated, heat oil to a depth of not more than 1/16 to 1/8 inch. You just need enough oil to cover the pan generously. In my All-Clad Stainless 12-Inch Fry Pan, that's 1/2 cup. When the oil is hot, put in two pieces at a time; don't crowd the pan. Cook for about 3 or 4 minutes, or until well browned and crunchy. With tongs, turn steaks and cook second side for about 3 or 4 minutes. Turn carefully so as not to lose the crust; chicken fried steak is all about the crust.