Heat 2 tablespoons of butter in a large skillet over medium heat. Add the peppers and onion, season with the salt and cook, stirring occasionally, until soft and golden, 8 to 10 minutes.
Throw the roast beef into the same skillet, breaking it up a little as you go. Add a few dashes of Worcestershire and warm for about a minute or so. Remove from the heat.
To build: Heat a panini maker (or set a grill pan or skillet over medium-low heat).
Spread half of the mustard onto a slice of bread and top with 2 slices of provolone. Add half of the beef and veggie mix. Top with half of the Homemade Cheese Sauce and another slice of bread. Spread the top side of the bread with 1 tablespoon of butter. Repeat and build the second sandwich.
Invert the sandwiches butter-side down on the hot panini maker (or grill pan or skillet). Spread another tablespoon of butter on each of the tops.
Close the panini maker and grill until the bread is crusty and golden and the cheese is melted, 4 to 5 minutes. (If using a grill pan or skillet, carefully flip the sandwiches halfway through.) Slice in half and enjoy!
Homemade Cheese Sauce:
Place a double boiler over medium heat. Add the cubed cheese and milk and heat, stirring, until the cheese is melted and the mixture is combined, about 5 minutes. Stir in the mustard, vinegar and salt. Grate the garlic into the mixture using a rasp grater; stir to combine. Cook until the flavors meld, an additional 3 to 5 minutes.