Trim any excess fat from the steak, season with salt and pepper and drizzle each side with 1/2 tablespoon balsamic vinegar. Set aside for 20 minutes to come to room temperature.

While the steak is marinating, heat the olive oil over medium heat. Add the onion and sauté, stirring frequently, until soft and translucent, 3 to 4 minutes. Add the mushrooms and continue cooking until the mushrooms are nicely browned and the onion is caramelized, 8 to 12 minutes longer. Season to taste with salt and pepper, then remove from the heat and set aside.

Preheat a gas or charcoal grill to a medium-high temperature (about 400°F). Oil the grates, add the steak and grill to desired doneness, 2-1/2 to 3 minutes per side for medium-rare (depending on thickness). Leave the grill on. Allow the steak to rest for 5 minutes, then cut into very thin (1/8-inch) slices.

To assemble the sandwiches, place the baguette on a sheet of heavy-duty aluminum foil. Line the bottom of the baguette with spinach leaves, add the steak slices and top with the onion-mushroom mixture. Add the blue cheese and place the sandwich on the grill until the cheese melts, 2 to 3 minutes. Top with the other half of the baguette, compress gently and cut into 4 equal portions.