Lightly butter the outsides of the buns and toast on the grill for 1 minute per side. Remove to a platter.
Grill the sausages, turning occasionally, for 10 to 15 minutes or according to the package instructions.
For the cheese sauce: Combine the processed cheese, milk, salt, pepper and 1 teaspoon Montreal seasoning in a pan and stir. Leave on a gentle heat so that the cheese melts and a thickened sauce develops, about 8 minutes.
For the dogs: Heat a cast-iron skillet over a high heat and add the olive oil. When the oil is hot, add the bell peppers, onions and mushrooms. Add the remaining teaspoon Montreal seasoning. Cook, stirring occasionally, until the vegetables blacken, about 6 minutes.
Open the buns and put the sausages inside, then top with the veggies. Spoon over the cheese sauce and dig in.