Preheat oven to 425 degrees Fahrenheit. Prepare your veggies, keeping in mind that you want them to be roughly the same size for even cooking. Remove the florets from the broccoli stems and slice them into bite-sized pieces. Slice the bell pepper into thin strips. Halve the onion and slice it into thin strips as well (no need to separate the strips of onion from each other).
Transfer the prepared veggies to a large, rimmed baking sheet. Drizzle on the olive oil and balsamic vinegar. Add the salt and toss until all of the ingredients are evenly combined. Arrange the veggies in a single layer on the baking sheet. Sprinkle lightly with red pepper flakes and a few twists’ worth of freshly ground black pepper. Bake until the veggies are tender and deeply caramelized, about 20 minutes, tossing halfway.
To prepare your sandwiches: Spread a very thin layer of Dijon mustard on one slice of bread. Top the mustard with a heavy sprinkling of shredded cheese, then as much roasted vegetables as you can reasonably fit on top. Top with more cheese and place another piece of bread on top. Repeat for the remaining three sandwiches.
To cook your sandwiches: Heat a large skillet, preferably cast iron, over medium heat. Once it’s hot, add a generous pat of butter. Let the butter melt and swirl it around. Carefully place one sandwich on one side of the pan, then another sandwich on the other side. Cover the pan with a lid or a baking pan to encourage the cheese to melt. Let the sandwich cook until the bottom side is golden and the cheese is mostly melted. Use a spatula to life one sandwich out of the pan at a time. Add a little more butter to the pan and let it melt. Carefully flip the sandwiches over and place them back in the pan to cook the other sides until they’re nice and golden (I covered the sandwiches again here). Once the cheese is melted through and both sides are golden, transfer the sandwiches to serving plates. Repeat with remaining sandwiches.