In a large skillet, heat oil over medium-high. Season both sides of steaks with salt and pepper. Cook steaks until medium-rare, about 5 minutes per side, flipping once. Transfer to a plate; tent loosely with foil to keep warm.
Add onion to pan; season with salt and pepper. Cook until onion begins to soften, about 4 minutes. Add mushrooms; season with salt and pepper. Cook until mushrooms brown, about 6 minutes. Add vinegar and any accumulated juices from steaks; cook until evaporated, about 30 seconds. Transfer mushroom mixture to a medium bowl. Add spinach to pan; season with salt and pepper. Cook, stirring, until wilted, about 3 minutes.
Thinly slice steaks against the grain. Spread mustard on 4 slices bread. Dividing evenly, top with sliced steak, mushroom mixture, and spinach. Top with remaining bread. Serve immediately.