Make the pulled pork: In a medium bowl, mix together the pepper, cumin, oregano, garlic, and lime juice. Set aside.
Make a few incisions, 1-2 inches (2-5 cm) deep, on both sides of the pork butt, cutting against the grain. Sprinkle generously with salt all over, make sure to season inside the incisions.
Transfer the pork butt to a roasting dish, and pour the marinade all over. Make sure the pork is well coated, including inside the incisions. Cover and marinate for 8-10 hours, or overnight.
Preheat the oven to 250˚F (120˚C).
Roast the pork, covered, for 8-10 hours, or until the pork is tender and easily falls apart when prodded with a fork or tongs.
Let cool enough to handle, then remove any bones, and shred the pork with tongs. Keep in the fridge up to 5 days.
Make the Cuban sandwich: Preheat the oven to 350˚F (180˚C). Line a baking sheet with parchment paper.
Set the bread open face on the prepared baking sheet and spread the mustard over one side. Top with the Swiss cheese. Brush the garlic oil on the other side of the bread. Top with the ham. Arrange the pulled pork over the cheese.
Toast in the oven for 2 minutes, or until the cheese melts. Add the pickles and press the sandwich.
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