Combine the garlic, cumin, pepper, oregano, orange juice, lime juice, olive oil, and salt in a bowl and whisk to combine.
Place the pork and marinade in a large Ziploc bag and refrigerate for at least two hours or overnight, flipping a few times.
To roast, preheat oven to 275°. Line a rimmed baking sheet with two layers of heavy-duty aluminum foil. Place pork plus some of the marinade on the pan and fold up the top layer of foil, crimping the edges together while leaving an air pocket inside the foil.
Place in oven and roast for 3 hours.
At the 3-hour mark, open the foil, and set the oven to 325°.
Roast for 2-3 more hours, basting on occasion until the meat is fall-apart tender. Remove pork from the oven and let the pork rest until it's cool enough to shred into bite-sized pieces, discarding excess fat.
To make sandwiches, cut the ciabatta in half lengthwise.
Mix together equal amounts of the mayonnaise and Dijon and spread on one half of the loaf.
Top with a layer of cheese, then sliced ham, shredded pork, pickles, and then more cheese. Top with the other half of the bread.
Cut into slices and grill on your panini maker or cook in a skillet weighing down the top of the sandwich with a cast iron pan or another weight.
Sandwiches are ready when the cheese is melted and the meat is warm.
