Combine the chicken broth, sour orange juice, garlic, oregano, cumin, olive oil, salt, and pepper in a saucepan. Bring to a gentle simmer over medium heat.

Add the thick-sliced ham and warm through, spooning the broth over occasionally, until the ham reaches at least 135°F and has visibly picked up color and aroma from the marinade — about 5–8 minutes. Don't let it boil hard; you're warming and flavoring the ham, not cooking it.

Remove ham from the broth and hold at or above 135°F on the line until service.

Whisk the sour orange juice, garlic, oregano, cumin, olive oil, salt, and pepper together in a bowl or resealable bag. Add the pork shoulder, turning to coat fully. Refrigerate at least 4 hours, preferably overnight.

Preheat oven to 325°F. Transfer the pork and all the marinade to a covered roasting pan. Roast until the pork pulls apart easily with a fork and reaches at least 195°F internally, about 3–3½ hours.

Rest the pork 15 minutes, then slice thin against the grain. Hold at or above 135°F on the line until service.

Split the Cuban bread and spread mustard edge-to-edge on both cut sides — get it all the way to the corners on the bottom half especially, since that's where it's doing the most work as a moisture barrier.

Layer on the bottom half: Swiss cheese, the cold sweet ham, the warm mojo pork or mojo ham, and salami if using. Top with dill pickles, then close the sandwich.

Butter the outside of both crusts lightly. Preheat the smooth press to 350°F and wait for the READY light before loading the sandwich.

Close the press fully and hold until the crust is deep golden and audibly crisp, 4–5 minutes.

Rest 1 minute, then slice on a sharp diagonal and serve.