In large bowl, whisk oil, orange zest, orange juice, lime juice, garlic, cilantro, mint, oregano, cumin, salt and pepper. Add pork. Turn until well coated all over. Transfer pork and marinade to large, resealable plastic bag. Place bag in bowl. Refrigerate, turning occasionally, 12 to 24 hours.
Discard marinade. Pork may be netted. If not, and if desired, tie with kitchen twine in 4 spots to form tight roll. Set pork, fat side up, on rack set in shallow roasting pan or on rimmed baking sheet.
Roast, uncovered, in preheated 425F (220C) oven 30 minutes; baste with pan juices. Reduce heat to 375F (190C). Roast, basting every half hour, until instant-read thermometer insert in centre registers 160F (71C), 90 to 120 minutes.
Transfer pork to cutting board. Cover loosely with foil. Let stand 30 minutes before discarding net or twine.
Slice as thinly as possible. You can also refrigerate until cold to make it easier to slice.
Heat panini/sandwich press over medium. Add ham and pork. Cook, turning once, until warmed and browned in spots, about 1 minute. Transfer to plate.
Generously butter cut sides of baguettes. Toast on press over medium until lightly browned, 1 to 2 minutes. Transfer to work surface. Layer bottom of buns with pork, ham, cheese and pickles. Brush inside of bun tops with mustard. Close sandwiches.
Generously butter outside of sandwiches. Place on press. Cook until cheese melts and outsides are browned and crispy, about 3 minutes.
If you don’t have a press, use a large, cast-iron skillet or griddle. Top sandwiches with a large baking sheet weighed down with heavy cans or cast-iron skillet. Cook over medium for about 3 minutes per side.
Let sandwiches stand 1 minute before cutting in half. Serve hot.