Preheat your oven to 375 degrees Fahrenheit and prepare a baking sheet or large baking dish by lining it with parchment paper.
Wash and dry the green peppers. Slice them in half from top to bottom and discard the seeds inside. Place the pepper halves open-side up on the baking sheet.
Once the oven reaches 375 degrees Fahrenheit, add the empty peppers to the oven for 10 minutes.
Heat a large skillet over medium heat and add the olive oil and butter.
When the butter has melted, add the sliced onions and cook for about 7-10 minutes, until the onions are very soft and have begun to caramelize.
Add the salt, pepper, thyme, sliced mushrooms and beef stock (or red wine, if using), and stir well.
Continue to cook for another 7-10 minutes until the mushrooms are soft and cooked through and the liquid has evaporated.
While the mushrooms and onions are cooking, slice the steak into thin strips (about 1/3 inch thick) and add them to a bowl.
Remove the baking sheet with the pepper halves from the oven after 10 minutes and set the baking sheet aside while you finish preparing the filling.
Once the mushrooms and onions are cooked and soft, add the mixture to the bowl with the steak, and 2/3 cup of the shredded cheese. Mix very well.
Divide the mixture evenly between the 8 pepper halves on the baking sheet and stuff the peppers with the steak mixture.
Top each one with the remaining cheese.
Bake for 15-20 minutes at 375 degrees Fahrenheit and then broil for about 1-2 minutes (depending on the strength of your broiler) to brown the cheese on top.
Serve and enjoy!