Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Season the steak with ½ teaspoon each salt and pepper.
Cook steak 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 oil over medium-high heat. Add the mushrooms, ½ teaspoon salt, and ¼ 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.