1 (20-ounce) piece beef tenderloin, cut from large end (head)
1 teaspoon olive oil
Coarse salt and freshly ground black pepper
Bearnaise Sauce, recipe follows
Heat oven to 250 degrees F.
Stand meat on its wider cut end on a solid work surface. Fold a kitchen towel once or twice, lengthwise, into a wide band and drape it loosely around the upright meat like a scarf. Gather the ends and twist to form a tight collar that will hold meat in a round shape during pounding; grasp securely in 1 hand. With the other hand, pound top of meat with a meat mallet, until flattened into an oval steak, about 2 inches thick and 5 to 7 inches wide.
Heat a medium grill pan over high heat to very hot. Rub steak with oil and season with salt and pepper.
Place steak on grill pan at an angle to ridges, and cook until clearly marked, 2 1/2 to 3 minutes. Turn, maintaining same angle and repeat on second side. Turn again, rotating so visible grill marks are perpendicular to ridges. Cook until a crisscross is clearly marked, 2 1/2 to 3 minutes. Turn once more, maintaining same angle, and grill so second side has been on grill exactly as long as first side. Brown steak on outside edge, holding with tongs or propping against a potato shifting meat so entire edge gets browned. Grill a total of 10 to 12 minutes, depending on thickness of steak and desired degree of doneness.
Meat will be barely springy to the touch, indicating that it is still rare inside. Transfer to a small shallow roasting pan and place in oven for 15 minutes. Remove to a cutting board, slice thinly on the bias. Place on a warm serving platter and pour pan juices over.
2 tablespoons tarragon vinegar
2 tablespoons dry white wine
1/4 cup very finely chopped shallots
1/4 teaspoon ground black pepper, more if desired
1 tablespoon finely chopped tarragon leaves
3 egg large yolks
6 to 8 ounces (1 to 1 1/2 sticks) unsalted butter, very soft
Pepper, if desired
Lemon juice, if desired
Combine vinegar, wine, shallots, black pepper, and 1 1/2 teaspoons tarragon in a small saucepan. Cook over medium heat until reduced to 1 tablespoon, 5 to 10 minutes.
Add egg yolks and 1 tablespoon water to reduced vinegar mixture. Whisk until thick and pale, about 2 minutes. Set pan over moderately low heat and continue to whisk at reasonable speed reaching all over bottom and insides of pan, where eggs tend to overcook. To moderate heat, frequently move pan off burner for a few seconds, then back on. As they cook, the eggs will become frothy and increase in volume, then thicken. When the bottom of the pan is visible in the streaks left by the whisk and the eggs are thick and smooth, remove from heat.
By spoonfuls, add soft butter, whisking constantly to incorporate each addition. As the emulsion forms, add butter in slightly larger amounts, always whisking until fully absorbed. Continue incorporating butter until sauce has thickened to consistency desired. Season with salt, remaining 1 1/2 teaspoons chopped tarragon and, if desired, pepper. Add a few droplets of lemon juice if necessary.