Preheat the oven to 400°F. Slice peppers in half and remove ribs and seeds. Line a baking sheet with parchment paper and lay each half on the sheet. Pop the peppers in the oven while preparing the filling, roughly 15 minutes or until desired tenderness.
Slice the onion into thin strips, and dice mushrooms if using. Heat a skillet over medium high heat. Add vegetables to pan and sauté.
As vegetables begin to cook down, add butter and garlic, and sauté until onions begin to become transparent.
Push the vegetables to the side, and add shaved beef to the pan. You can add a splash of oil or butter to the pan as needed. Season with salt and pepper (or other seasonings as desired).
Since the beef typically comes in sheets, I'll usually do one or two at a time and push them over with the veggies when I add the next. Once the beef is cooked through toss together with the vegetables and remove from the heat. Optionally, use a paper towel to soak up some of the grease.
Remove peppers from the oven, and blot or pour out any moisture that's collected from roasting them. Stuff each pepper half full of filling, and top with cheese.
Broil for 30 seconds or until cheese is golden and bubbly. Serve warm.