Cut the meat with the grain into 1/2 inch thick slices. Season each piece with salt and pepper. Place the flour in a pie pan and the eggs in a separate pie pan. Dredge the meat on both sides in the flour. Tenderize the meat into 1/4 inch thick slices. Dredge the meat again in the flour, followed by the egg, and then back in the flour. Repeat with all the meat. Allow to sit 10-15 minutes.
Place enough vegetable oil in a skillet to cover the bottom and set over medium high heat. Once the oil is shimmering, add the meat in batches, not overcrowding the pan. Cook each side until golden brown (about 4 minutes per side.) Remove the steaks to a wire rack set in a half sheet pan and place into the oven.
Add the remaining oil to the pan. Whisk 3 tablespoons of the remaining flour left over from dredging. Add the chicken broth to deglaze the pan. Whisk until the gravy until it boils and begins to thicken. Add the milk and whisk until the gravy coats the back of a spoon (5-10 minutes.)
