2) Cut the meat with the grain into 1-cm thick slices. Season each piece on both sides with the salt and pepper. Place the flour into a pie pan. Place the eggs into a separate pie pan. Dredge the meat on both sides in the flour. Tenderize the meat, using a needling device, until each slice is 1/2-cm thick. Once tenderized, dredge the meat again in the flour, followed by the egg and finally in the flour again. Repeat with all the pieces of meat. Place the meat onto a plate and allow it to sit for 10 to 15 minutes before cooking.
3) Place enough of the vegetable oil to cover the bottom of a 30-cm slope-sided skillet and set over medium-high heat. Once the oil begins to shimmer, add the meat in batches, being careful not to overcrowd the pan. Cook each piece on both sides until golden brown, approximately 4 minutes per side. Remove the steaks to a wire rack set in a half sheet pan and place into the oven. Repeat until all of the meat is browned.
4) Add the remaining vegetable oil, or at least 1 tablespoon, to the pan. Whisk in 3 tablespoons of the flour left over from the dredging. Add the chicken broth and deglaze the pan. Whisk until the gravy comes to a boil and begins to thicken. Add the milk and thyme and whisk until the gravy coats the back of a spoon, approximately 5 to 10 minutes. Season to taste, with more salt and pepper, if needed. Serve the gravy over the steaks.