Preheat the oven to 400°F.

Slice about a third off the top of each pepper, from stem to end, and remove the ribs and seeds inside the pepper.

Slice the pepper tops into thin strips for later use.

Place the peppers cut side-up in a baking dish and bake for 30 minutes.

In a large skillet or on a griddle over medium heat, heat 2 tablespoons olive oil. Add the reserved sliced pepper tops, onions, mushrooms, 1/2 teaspoon salt, and 1/2 teaspoon black pepper. Cook until softened and caramelized, about 12 to 15 minutes.

In the same skillet or griddle over medium-high heat, heat the remaining 1 tablespoon olive oil. Add the sliced steak and the remaining 1/2 teaspoon salt and 1/4 teaspoon black pepper. Cook for 1 to 2 minutes without stirring, then give a quick stir to finish cooking.

Transfer the cooked steak to a cutting board and chop it roughly, then combine with the cooked vegetables.

Remove the baked peppers from the oven. Fill each pepper half-full with the filling, place a slice of provolone on top, and fill to the top with more filling. Halve the remaining 3 slices of cheese and top each pepper with them. Sprinkle with thyme.

Bake the stuffed peppers for 15 minutes until the cheese is melted and the peppers are tender. Serve warm.