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Heat 2 tablespoons oil in a large heavy skillet over medium-high heat. Add mushrooms; cook, stirring occasionally, until soft and golden, about 7 minutes. Season to taste with salt and pepper. Transfer to a bowl; set aside.
Melt 1 tablespoon butter with remaining 1 tablespoon oil in same skillet over medium heat. Season steak with salt and cracked pepper. Add steak, garlic, and rosemary to skillet. Cook about 3 minutes per side for medium-rare. Transfer to a cutting board. Let rest while preparing sauce.
Discard garlic and rosemary from skillet. Pour off all but 1 tablespoon fat. Add wine; cook, stirring up bits, until reduced to ¾ cup, about 3 minutes. Strain; return liquid to skillet. Stir in stock; bring to a boil. Simmer until reduced to ½ cup, about 5 minutes. Remove from heat; Whisk in 3 tablespoons butter. Stir in mushrooms and 1 tablespoon tarragon. Season with salt and pepper.
Spoon mushroom mixture onto plates. Thinly slice steak; serve over mushrooms. Garnish with remaining 1 tablespoon tarragon.