Combine the olives, giardiniera, pepperoncini, cocktail onions, garlic, oil, and oregano in a food processor. Pulse until finely chopped.
Slice each piece of bread in half horizontally. Divide the olive relish into 4 portions. Spoon 1 portion on the bottom of each bread, about ½ cup each.
Layer the mortadella, soppressata, provolone, and ham on top of the olive relish. Spoon the remaining 2 portions of relish on top of the ham.
Top each sandwich with the top piece of bread and press down firmly. Slice each sandwich into quarters to serve.