Season the steaks with salt and pepper. Place 3/4 cup flour in a shallow bowl. In another shallow bowl, beat the eggs. Coat the steaks with the flour, then the egg, letting any excess drip off. Coat again with the flour.
In a large, heavy skillet, heat 1/4 cup oil over medium heat. Add 2 steaks and cook, turning once, until golden, 5 to 6 minutes. Drain on paper towels. Wipe out the pan. Repeat with the remaining 1/4 cup oil and 2 steaks.
Pour off the excess oil from the pan. Add the butter and melt over medium heat. Whisk in the remaining 1/4 cup flour for 1 minute. Gradually whisk in the chicken broth and milk until thickened, about 2 minutes. Season with salt and pepper, then spoon over the steaks.