Preheat toaster oven to 300 degrees F (150 degrees C).
Heat 1 1/2 teaspoon butter and 1 1/2 teaspoon oil in a large nonstick skillet over medium-high heat until hot but not smoking, about 2 minutes. Toss in mushroom slices; season with thyme leaves, salt, and pepper. Cook and stir until mushrooms start to brown and are tender, about 5 minutes. Drizzle soy sauce over mushrooms, toss briefly and transfer to a bowl.
Wipe out skillet. Heat remaining butter and oil over medium-high heat until hot but not smoking, about 2 minutes. Toss in peppers and onions; season with salt and pepper. Cook and stir until soft, about 5 minutes. Transfer peppers and onions to the bowl with the cooked mushrooms; toss to combine.
Place rolls on toaster oven baking tray. Spread mustard over 2 halves. Place 1 piece provolone cheese over each half. Toast open-faced until cheese is melted, about 3 minutes.
Place the toasted rolls on plates and top with mushroom mixture.