Make the gravy: Melt the butter with the vegetable oil in a saucepan over medium heat. Sprinkle in the flour and cook, whisking, until smooth and deep golden brown, about 2 minutes. Gradually whisk in the milk and simmer, whisking to remove any lumps, until the gravy thickens, 4 to 6 minutes. Add 2 to 3 teaspoons salt and 1 teaspoon pepper. Cover and remove from the heat.
Make the steak: Place the steaks between 2 pieces of heavy-duty plastic wrap and pound to 1/8 inch thick with a mallet or heavy skillet. Season generously with salt and pepper. Whisk the flour, baking powder and 1 teaspoon each salt and pepper in a shallow bowl. In another shallow bowl, whisk the milk and eggs. One at a time, dredge each steak in the flour mixture, dip in the egg mixture, then return to the flour mixture, pressing firmly to coat. Transfer the breaded steaks to a rack.
Preheat the oven to 250 degrees F. Set another rack on a rimmed baking sheet. Heat about 1 inch of vegetable oil in a cast-iron skillet or Dutch oven over high heat until a deep-fry thermometer registers 350 degrees F. One at a time, fry the steaks until golden brown, turning once, about 5 minutes (return the oil to 350 degrees F between each steak). Transfer the finished steaks to the rack on the baking sheet and keep warm in the oven while you make the rest.
Meanwhile, reheat the gravy. Stir in some water, 1 tablespoon at a time, if the gravy is too thick. Drizzle the fried steak with the gravy and top with scallions.