In a large dutch oven, heat oil over medium heat. Add some of the stewing beef to the skillet and brown all sides. Place just enough beef cubes to cover the bottom of the dutch oven. Once browned, remove the beef and add some more. Continue to brown the beef in batches until they are all browned. Remove beef and set aside in a large bowl.
Melt butter in the dutch oven. Add the onions, green and red pepper, mushrooms and minced garlic. Saute, stirring often, until the vegetables are tender, about 6-8 minutes. Stir in the beer, tomato paste, paprika, salt and pepper. Bring to a boil, lower heat and let simmer for 3-5 minutes. Remove from the heat and mix in the brown stewing beef.
Put a lid on the dutch oven and bake in the oven for about 3 hours. Remove from the oven and increase the oven temperature to broil.
Slice a crusty bun lengthwise, making sure to not slice all the way through. Open it up and place on a large baking sheet lined with aluminum foil. Spoon some of the beef and vegetable mixture in the opening of the crusty bun. Spoon a little extra sauce on the bun. Sprinkle some grated cheese on top of the beef mixture. Do this with the remaining crusty buns.
Place in the oven and broil until the cheese melts, about 3-5 minutes.