This sandwich is named for the trio of meats that come on it (typically grilled steak, lechon and ham). It was created by take-out joints as a late-night snack for the bar and club crowd in Puerto Rico, and Axel Perez is carrying on the tradition at his spot by selling tripletas outside a local nightclub every weekend. His have a Texan twist: nacho cheese in place of the usual Swiss or muenster.
Combine the salt, oregano, pepper and cumin in a bowl. In a blender, combine the vegetable oil and garlic and blend until smooth. Stir 1 cup of the garlic oil into the seasoning mix to make the marinade. (Set aside the remaining garlic oil.)
Put the pork on a rimmed baking sheet and coat with some of the marinade, using about 1 tablespoon for each side. Refrigerate for at least 4 hours.
Toss the beef with 1/4 cup of the marinade in a bowl, then refrigerate for at least 4 hours.
Preheat the oven to 350˚ F. Put the pork on a clean baking sheet, cover with foil and roast until the pork is soft and very moist, 4 to 5 hours. Remove from the oven and let rest for 15 minutes, then shred the pork.
After placing the pork in the oven, put the beef in a Dutch oven and add enough water to cover. Add the remaining 1 cup garlic oil and cook over medium heat until the beef is soft, about 2 hours. Remove from the heat and drain.
Heat a griddle over medium-high heat and coat with cooking spray. Add some of the beef to the griddle, then some pork and then some ham; cook, breaking up the meat and stirring everything together with a spatula, until heated through, 3 to 4 minutes. Repeat to cook the remaining meat.
Spread the cut sides of the rolls with mayonnaise and cheese sauce. Pile on the meat; top with the potato sticks and ketchup. Close the sandwiches and cut in half.