If you want to get fancy, you can call it "Beef Paillard", and prepare it using filet mignon; but that really isn't necessary. Plain old Chicken Fried Steak is a down-home, classic American favorite. Chicken Fried Steak is now popular almost everywhere, but it is most commonly associated with Texas, where it is something akin to a State Dish. That the Texas-Chicken Fried Steak link is so strong is evidenced by that fact that a database search of Texas Monthly magazine for the last several years returns 106 hits for the term, and not just in food related articles!
A distinction is sometimes made between Chicken Fried- and Country Fried Steak. If there is a categorizable difference, it might be that the latter, which is perhaps more popular in the Southeast, is "smothered" - being returned to the pan after the gravy is prepared, and served completely moistened with gravy - whereas the former, preferred in Texas, is served "dry", with some gravy poured on top, and on the side.
The dish probably finds its ultimate origins in central Europe; think Wienerschnitzel. In the 19th century, German immigrants populated the Texas hill country, and brought a hankering for this type of dish. Instead of the veal of Wienerschnitzel, a thinly sliced or pounded cheap cut of beef, usually round steak, is used. A gravy is made from the pan drippings; mashed potatoes make the perfect accompaniment.
This recipe is adapted from one by Threadgill's in Austin, TX
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1 large egg
2 cups milk, divided
3/4 cup flour
4 Six ounce sirloin or round steaks, pounded to about 1/4 inch thickness
Vegetable or Olive Oil
1 Tbsp flour
1 tsp Worcestershire sauce
1/2 tsp hot sauce
Salt and Pepper
Whisk the egg and 1 cup milk together in a medium bowl, set aside.
Combine the 3/4 cups flour, seasoned with 1/2 tsp each of salt and pepper in a medium bowl.
Dip the steaks in the egg mixture, and dredge in the flour mixture.
Re-dip the steaks in the egg mixture, and immediately fry them, in batches, in about 3/4 inch of oil in an iron skillet, over high heat for about 2 minutes per side, until golden. Set the steaks aside to drain on paper towels; keep warm.
Reserve about a tablespoon of drippings in the skillet. Heat and whisk in the Tbsp of flour, cook until you have a golden roux. Add the remaining cup of milk; cook, stirring, until the gravy is thick and bubbly. Add the Worcestershire sauce, hot sauce, and season with salt and pepper to taste. Serve the gravy with the steak.