Slice crusts off top and bottom of each ciabatta loaf, leaving the sides. Cut loaves in half horizontally. Spread a thin layer of mayonnaise and mustard on bread. Layer half of 1 loaf with a slice of cheese, a thin layer of pork, a layer of peperoncini, a second slice of cheese, 2 slices of ham, a layer of pickles, and a third slice of cheese. Top with remaining half of loaf, spread side down. Repeat.
Preheat a grill pan over medium-high heat. Place panini in pan. Place a heavy pan on top, pressing down, and cook until browned, about 4 minutes. Flip, pressing down, and cook until browned, about 4 minutes more. Halve sandwiches; serve immediately.