Preheat the broiler. Combine the fig preserves and mustard in a small bowl; set aside. Hold the Brussels sprouts by the stem ends and grate them on the large holes of a box grater. Transfer to a foil-lined baking sheet and toss with the olive oil; season with salt and pepper. Broil until lightly golden brown and just starting to crisp, 3 to 4 minutes.
Spread 1 tablespoon of the fig mustard on the bottom halves of each roll. Top each with 1 cheese slice, some Brussels sprouts, 3 or 4 salami slices, another cheese slice and the roll top.
Melt 1 tablespoon butter in a large skillet over medium-low heat. Add 2 sandwiches, then press down with a heavy skillet and cook until golden brown, 5 to 6 minutes. Flip the sandwiches, adding 1 more tablespoon butter to the skillet; press down again and cook until browned and the cheese is melted, 4 to 5 minutes. Wipe the skillet clean and repeat with the remaining 2 sandwiches and butter. Serve with vegetable chips.