Set a rack in the center of the oven and heat the oven to 400°F. Season the steak generously with salt and pepper. Heat an ovenproof 12-inch sauté pan over high heat until very hot. Add the oil to coat the pan and sear the steak on one side until well browned, about 1-1/2 minutes. Flip with tongs and cook until the second side is well browned, about another 2 minutes. Put the pan in the oven and roast until the steak is cooked to your liking, 5 to 7 minutes for medium rare. Transfer the steak to a cutting board and let rest, lightly covered with foil, while you make the sauce.
Set the same skillet over medium heat and add 2 Tbs. of the butter. When the butter melts, add the shallot and garlic and cook until soft and translucent but not browned, about 2 minutes. Add the wine, increase the heat to medium high, and boil until syrupy, 2 to 4 minutes. Add the broth and thyme and boil until about 1/3 cup of liquid remains, 6 to 8 minutes. Reduce the heat to low and whisk in the mustard. (Don’t let the sauce boil after the mustard is added.) Stir in the 4 Tbs. butter cubes, a few at a time. Stir in half the tarragon and season the sauce to taste with salt and pepper.
With a sharp knife, slice the steak thinly across the grain on the diagonal, drizzle with the sauce, and sprinkle with remaining tarragon.