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Heat the oil in a nonstick saute or frying pan over medium heat. Pan-fry 2 steaks at a time, 1 to 1 1/2 minutes per side. Drain on paper towels. Keep warm.
Making the peppered gravy: Drain excess oil from the pan, leaving about 2 tablespoons. Add the flour and stir to blend. Cook, while stirring, until the flour smells nutty and begins to brown, about 2 minutes. Whisk in the beef stock, balsamic vinegar, black pepper and cayenne. Bring to a boil, then reduce the heat to low. Simmer the gravy until it is thick enough to coat a spoon, 3 to 5 minutes. Season with salt and pepper, stir to blend, and keep warm.
Plate the chicken-fried steaks and smother with gravy. Serve with bitter greens and mashed potatoes.