In a large skillet, heat butter and olive oil over medium heat.
Add onions and cook for 8–10 minutes until caramelized.
Stir in mushrooms, garlic, thyme, soy sauce, and black pepper. Cook until mushrooms are golden and tender, about 8 minutes.
Preheat oven broiler. Place hoagie rolls on a baking sheet, cut side up. Fill each with mushroom mixture and top with 2 slices of cheese.
Broil 1–2 minutes until cheese is melted and bubbly.
Meanwhile, in a small saucepan, combine vegetable broth, Worcestershire sauce, soy sauce, onion powder, and garlic powder. Bring to a simmer for 5 minutes.
Serve sandwiches hot with au jus on the side for dipping.