Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Season the steak with 1/2 teaspoon each salt and pepper. Cook to the desired doneness, 3 to 5 minutes per side for medium-rare. Let rest at least 5 minutes before slicing.
Wipe out the skillet and heat the remaining tablespoon of oil over medium-high heat. Add the mushrooms and season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook, tossing occasionally, until tender and browned, 5 to 6 minutes. Stir in the vinegar. Remove from heat and stir in the cream.
Divide the arugula, steak, and mushrooms among plates and sprinkle with the scallions.
Substitution: You can use almost any variety of mushroom in this recipe. Try smoky shiitakes (discard the woody stems), mild oysters, or a couple of juicy portobellos.