Preheat oven to 375°F. Line a baking sheet with parchment paper.
In a small bowl, combine finely chopped olives and giardiniera. Set aside.
In a small bowl, combine beaten egg with 1 teaspoon water to create an egg wash. Set aside.
Cut the block of mozzarella in half across the middle, and cut each half into 4 rectangles or sticks (or use 1 ounce mozzarella sticks). Wrap each stick in a slice of mortadella, so that it is completely covered. Set aside.
Lay 2 sheets of phyllo dough at a time on a clean surface. Brush one gently and quickly with melted butter. Place the other sheet on top. Save remaining sheets under a slightly damp paper towel and cover with plastic while working so that they do not dry out.
To assemble, lay 2 slices of salami centered on the left side of the buttered phyllo sheet, leaving a 2-inch border on the left. Place a rounded tablespoon of the olive mixture on top of the salami. Place the mortadella/mozzarella package on top. To roll, fold the 2-inch border over top of the entire filling. Tuck in the top and bottom sides, and continue to roll the rest of the way toward the right side. Brush butter along the last couple inches to seal, and make sure that the seam is underneath the roll. Repeat for all 8 rolls. Assemble one roll at a time completely before moving on to the next.
Place each roll on a baking sheet at least 2 inches apart. Brush with remaining butter.
Brush tops with egg wash and sprinkle with sesame seeds and Parmesan.
Rolls can be chilled at this step and baked fresh to order or baked and reheated to order.
Bake for 18 to 20 minutes or until deep golden brown.
Let cool for 5 minutes and serve warm.
To reheat individually to order: Bake for 4 to 5 minutes at 350°F or until warmed through and crispy.
To plate: Slice each roll in half on the diagonal. Stack rolls on plate and garnish with additional olive mixture and fresh parsley on the side, if desired.
