Make the mojo: In a bowl, mash the garlic with the kosher salt, then stir in the ground cumin, dried oregano, and black pepper. Whisk in the orange juice, lime juice, and olive oil, then add the bay leaves.
Marinate the pork: Place the pork shoulder in a nonreactive dish or large zip-top bag and pour the mojo over it, turning to coat. Cover and refrigerate 4–12 hours (8 hours ideal), turning once halfway through.
Roast the pork: Preheat the oven to 325°F. Remove the pork from the marinade, wiping off excess and discarding the bay leaves and marinade. Set the pork in a small roasting pan and roast until the center reaches 155–165°F and the exterior is lightly browned, 1 hour 45 minutes to 2 hours 30 minutes; turn once for even cooking. Rest 15 minutes, then slice the pork very thinly across the grain.
Build the sandwiches: Lay the split Cuban bread open. Spread yellow mustard on the cut sides. Layer dill pickles on the bottom halves, then arrange a generous layer of sliced roast pork, followed by baked ham and Swiss cheese. Close the sandwiches.
Butter and press: Lightly spread the outsides of each sandwich with the softened unsalted butter. Heat a plancha, panini press, or large skillet over medium-low. Press the sandwiches, using a press or a heavy skillet to weight them, until the bread is crisp and deep golden and the cheese is melted, 3–5 minutes per side; the sandwiches should compress to about one-third of their original height.
Serve: Cut each sandwich on the diagonal and serve hot while the bread is crackly and the cheese is molten.