Place 1 cup flour in a shallow bowl. In a separate shallow bowl, whisk eggs and 1/2 cup milk until blended. Sprinkle steaks with 3/4 teaspoon each salt and pepper. Dip in flour to coat both sides; shake off excess. Dip in egg mixture, then again in flour., In a large cast-iron or other heavy skillet, heat 1/4 in. of oil over medium heat. Add steaks; cook until golden brown and a thermometer reads 160°, 4-6 minutes on each side. Remove from pan; drain on paper towels. Keep warm., Remove all but 2 tablespoons oil from pan. Stir in the remaining 1/4 cup flour, 1/2 teaspoon salt and 1/4 teaspoon pepper until smooth; cook and stir over medium heat until golden brown, 3-4 minutes. Gradually whisk in water and remaining milk. Bring to a boil, stirring constantly; cook and stir until thickened, 1-2 minutes. Serve with steaks.