Melt butter in a large skillet over medium heat. Add onions and cook, stirring occasionally, until they begin to soften, about 10 minutes.

Add brown sugar and balsamic vinegar to the onions. Continue to cook, stirring occasionally, until the onions are caramelized, about 15 more minutes. Remove from heat and set aside.

Season the steak slices with salt and pepper. Heat olive oil in a large skillet over medium-high heat. Add the steak slices and cook until browned and cooked to your desired doneness, about 3-4 minutes per side. Remove from heat.

Preheat the oven to 350°F. Slice the hoagie rolls in half and place them on a baking sheet. Layer each roll with steak slices, caramelized onions, and 2 slices of provolone cheese.

Place the sandwiches in the oven and bake until the cheese is melted and the rolls are toasted, about 5-7 minutes.

Remove from the oven and top each sandwich with fresh arugula. Serve immediately.