In a food processor, pulse the giardiniera and olives into a finely chopped relish.
Heat a large skillet over medium to medium-high. Lightly butter the rolls. Press the cut sides onto the pan until evenly browned, 4 to 5 minutes. Add the mortadella to the skillet a few slices at a time and cook, turning with tongs, until lightly browned, about 2 minutes per side.
Top the roll bottoms with some relish. Add a few slices of mortadella, a slice of cheese, more mortadella, then the remaining slices of cheese. Add the bun tops. In the same skillet, melt the remaining butter over medium. When the butter foams, add the sandwiches and cook until the cheese melts, pressing down on the rolls until deep golden brown and crispy, about 3 minutes per side.