Preheat the oven to 400.
Melt 2 tablespoons butter in a medium sauce pan and saute the peppers, onions and mushrooms until they are very soft--about 12-15 minutes.
Add salt and pepper to taste.
Transfer the vegetables to a plate on the side and cover with foil to keep warm.
Slice the roast beef into thin strips. It should already be shaved, so just cut it up a little.
Over medium heat, add the remaining butter and melt.
Add the roast beef and toss. Heat thoroughly. About 5 minutes.
Add the hot meat to the vegetables and toss to combine.
Place about 1/4 of the meat and veggie mixture back in the pan, scooting it to one side. Add another 1/4 to the other side.
Top each pile with one slice of cheese and allow it to melt slightly.
Transfer each pile on to its own hoagie roll and roll each up tightly in foil.
Complete the other two sandwiches then transfer to a hot oven for 7-8 minutes and you're ready to serve.
For the sauce:
Mix all ingredients well in a small bowl. Serve on the side for dipping.