For the mushrooms, in a large bowl or a zip-top bag, whisk together the olive oil, Worcestershire sauce, salt, pepper, garlic powder and lemon juice. Add the mushrooms and toss to combine.
Marinate for 30 minutes to 1 hour.
While the mushrooms marinate, make the onions: In a large skillet over medium-high heat, heat the olive oil. Add the onions and sauté until they begin to get tender, about 5 minutes. Season with salt and pepper, and reduce the heat to low. Cook, stirring occasionally, until the onions become a deep brown, caramelized color, about 20 minutes. If they start to look dry or stick to the pan, add a tablespoon or two of water.
For the blue cheese spread, mix the blue cheese and mayonnaise together in a small bowl.
Heat a grill pan over medium-high heat. Remove the mushrooms from the marinade and transfer to the grill pan. Cook until slightly wilted, browned and showing grill marks, 2-3 minutes per side.
Lightly toast the hoagie rolls on the grill pan then spread one side of each roll with the blue cheese spread. Top with mushrooms, caramelized onions and arugula. Drizzle lightly with balsamic vinegar, sprinkle with salt and serve.