Heat 3 tablespoons of the olive oil in a large nonstick skillet. Add the mushrooms and shallot, season with salt and pepper, and cook without stirring over moderately high heat until the mushrooms are softened and just beginning to brown, about 5 minutes. Continue cooking, stirring, until the mushrooms are tender and browned; transfer to a bowl.

Season the steaks generously with salt and pepper. Add the remaining 1 tablespoon of olive oil to the skillet and heat until almost smoking. Add the steaks and cook over moderately high heat, turning once, until well browned outside and medium rare within, about 5 minutes total. Transfer the steaks to a platter and let rest for at least 5 minutes.

Pour off the fat in the pan. Add the red wine to the skillet and boil, stirring frequently and scraping up any browned bits, until it has reduced by half, about 3 minutes. Stir in the barbecue sauce and simmer until the pan sauce is glossy, about 1 minute. Remove from the heat and whisk in the butter. Season with salt and pepper.

Slice the steak against the grain on the diagonal and arrange on plates. Top with the sauce and garnish with the scallions or chives. Serve the sautéed mushrooms on the side.