Cut the steak into 4 equal pieces. Season the steaks with 1 teaspoon salt and ¼ teaspoon pepper. In a shallow bowl, combine the 1¼ cups (6½ oz/200 g) flour with ½ teaspoon each salt and pepper. In a second shallow bowl, whisk together ½ cup (4 fl oz/125 ml) of the milk and the eggs. One at a time, coat the steak pieces with the flour mixture, shaking off the excess, then dip into the egg mixture, coating evenly and allowing the excess to drip off. Return to the flour mixture to coat evenly, again shaking off the excess. Transfer the coated steak pieces to a plate. 

Preheat the oven to 200° F (95° C). Set a wire rack on a rimmed baking sheet and place near the stove. Pour canola oil to a depth of about 1 inch (2.5 cm) into a large, heavy frying pan (preferably cast iron), and heat over medium-high heat until the oil shimmers. Add the steaks and cook until the undersides are golden brown, about 1½ minutes, then turn and cook until the other sides are browned, about 1½ minutes more. Transfer to the rack and keep warm in the oven while making the gravy. 

Carefully pour the hot fat from the frying pan into a heat-proof bowl and set aside. Wipe out the pan with paper towels. In a saucepan, heat the remaining 2 cups (16 fl oz/500 ml) milk over medium heat to just below a boil and remove from the heat. Return 2 tablespoons of the fat to the pan and heat over medium-low heat. Whisk in the remaining 2 tablespoons flour and let bubble gently for 2 minutes. Gradually whisk in the hot milk and bring to a boil over high heat. Reduce the heat to medium-low and simmer, whisking frequently, until lightly thickened, about 3 minutes. Season with salt and pepper and pour into a sauceboat. Serve the steaks at once, passing the gravy on the side