Season steak with salt and pepper on both sides.

Heat a heavy skillet over high heat until a drop of water immediately sizzles and evaporates when flicked onto the skillet's surface. Pour in oil. Add steak and cook until it is beginning to firm and is hot and slightly pink in the center, about 4 minutes per side. An instant-read thermometer inserted into the center should read 140 degrees F (60 degrees C).

Remove steak from skillet and set aside. Tent with foil to keep warm. Discard oil remaining in the skillet.

Reduce heat to medium-low and cook mushrooms, shallot, and garlic, stirring frequently, until softened, about 5 minutes. Pour in white wine and cook until slightly reduced. Pour in beef broth. Reduce heat to low and stir in butter, 1 cube at a time, stirring well after each addition. Stir in tarragon. Season with salt and pepper. Serve mushroom sauce over the steak.