Heat your panini press. If there are temperature settings, go for full power.
While the press is heating, prepare your ingredients by pre-stacking the fillings on a cutting board or a parchment paper-lined countertop. Remove 8 pickles from the jar and blot with paper towels, drying them as much as possible. Divide the ingredients evenly among the 4 rolls: Start with 2 slices ham, followed by one half-piece of cheese, 2 slices of pork or turkey, 2 pickles, finishing with another half-piece of cheese.
Halve the rolls lengthwise and brush the cut sides liberally with the melted butter. Place 1 or 2 rolls (or however many will fit comfortably) on the panini press, cut-side down, then coat the outside of the rolls with additional butter. Close the press and brown for 2 1/2 minutes. Repeat with remaining rolls.
While the rolls are in the panini press, tear four sheets of heavy-duty aluminum foil, enough to wrap each sandwich.
Remove rolls from the panini press and place on aluminum foil. Coat both cut sides of the rolls with enough mustard to cover. Position the pre-stacked ingredients on the bottom roll, finishing with the top roll. Wrap in foil and set aside. Build the second sandwich and place both, still wrapped in foil, back in the press for 3 minutes. If your panini press only holds one sandwich, just toast one at a time.
Rest for 1 minute in the foil, then unwrap and enjoy.