Make the beef tenderloin
1. Using a sharp, thin-bladed knife, remove the chain muscle from the beef tenderloin. (The chain muscle is the long, thin muscle, connective tissue, and fat that runs almost the length of the beef tenderloin.) Reserve it for another use, such as stir-frying or ground beef. Then remove the silver skin and most of the surface fat from the tenderloin.
2. Next, about 5 to 6 inches from the narrow end of the beef tenderloin, make a shallow cut crosswise across the tenderloin, cutting about halfway through the meat. Fold this narrow end piece under so that the tenderloin is uniformly thick from one end to the other. Secure the tenderloin with kitchen string, tying it at 1 1/2-inch intervals along the length of the tenderloin.
3. Place the tenderloin on a platter. Brush or rub the meat on all sides with 2 tablespoons of the olive oil. Season generously with salt and pepper, firmly pressing the seasonings in with your fingertips. (The roast can be refrigerated, uncovered, for up to 24 hours. Let stand at room temperature for about 1 hour before roasting.)
4. In a large skillet over low heat, warm the remaining 2 tablespoons olive oil. Add the onions, stir to coat with the oil, and season to taste with salt and pepper. Cook for 2 to 3 minutes, then add 1/4 cup water, the thyme, and the balsamic vinegar. Cover and cook over low heat, stirring occasionally, until the onions are very soft but not browned, about 15 minutes. Uncover and, if any liquid remains in the pan, increase the heat to medium-high and cook just until the liquid evaporates. Again, do not let the onions color.
5. Position a rack in the upper third of the oven. Preheat the oven to 425°F (220°C). Arrange the onions along the center of a shallow roasting pan just large enough to hold the tenderloin. (If you have a rather large roast, you can spread the onions on the diagonal to make space for it in the pan). Place the tenderloin on the onions.
6. Roast the tenderloin until a thermometer inserted into the thickest part registers 125°F (52°C) for medium-rare, about 45 minutes. The roast will vary somewhat in doneness depending on the thickness of its sections. Remove the pan from the oven, transfer the meat to a platter, and tent with aluminum foil. Let rest for 10 to 15 minutes before slicing.
Make the Madeira sauce
7. In a saucepan over medium-high heat, bring the 1/2 cup Madeira to a boil and cook until reduced to 1/4 cup.
8. Pour about 1/2 cup of the pan juices from the roasting pan through a fine-mesh sieve into the saucepan. Discard the thyme but reserve the onions in the roasting pan. Add the beef stock or broth to the saucepan, bring to a boil, and cook for 2 to 3 minutes to reduce slightly and blend the flavors. Reduce the heat to medium-low and stir in the demi-glace and red wine vinegar. Season to taste with salt and pepper and, if desired, 2 to 3 tablespoons more Madeira.
9. Meanwhile, place the butter and flour in a small bowl and mix together with a fork until well combined to make a beurre manié. [Editor's Note: A beurre manié is a simple manner of thickening a sauce, lending it a rich, smooth, velvety consistency.] Reduce the heat under the sauce to low, whisk the beurre manié into the liquid a little at a time, and simmer until the sauce reaches the desired consistency, 2 to 3 minutes. Keep the sauce warm by covering it and placing it over very low heat.
10. Transfer the tenderloin to a carving board. Remove the strings and carve the roast into slices about 1/2 inch thick. Place a spoonful of the reserved onions onto individual plates and top with a slice of tenderloin. Spoon a little of the sauce over the meat and pass the remaining sauce at the table.
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