Cuban Roast Pork Cheeseburger with Mojo Onions

For the pork shoulder: To make the pork shoulder, score the meat all over by making 1/2-inch deep cuts through the flesh on all sides. Place pork shoulder in a resealable plastic bag. Blend the remaining ingredients in a blender until smooth. Pour marinade over pork, seal the bag and let pork marinate overnight in the refrigerator. 

Heat oven to 250°F. Remove pork from the marinade (reserve marinade in the refrigerator until ready to use) and place, fat-side-up in a large, oven-safe frying pan, let sit out at room temperature while oven heats. Place pan in oven, add 1 cup of water and roast, uncovered, occasionally basting meat with juices. Cook for 6 hours or until meat shreds easily. If you are making the burger right away, you can let the pork rest until cool enough to handle then shred and keep warm. If you are making the pork in advance, shred the meat, cover and refrigerate until ready to use.

For the burgers: To make the burgers, form beef into six patties. Season generously with salt and pepper and coat with 1 tablespoon of the oil; set aside. Heat a medium frying pan over medium heat and add onions and reserved marinade. Cook until onions are soft and caramelized, about 15 to 20 minutes. If onions are getting too dark, lower the temperature a bit. 

Meanwhile mix together mayonnaise and mustard in a small bowl and toast the buns. When onions are ready, heat a large cast iron skillet over medium heat. Place burgers in pan (you may have to do in batches) and cook for 5 minutes a side for medium.

Turn off heat and top each burger with about 2 to 3 tablespoons of onions and a slice of cheese. Cover and let sit until cheese melts, about another minute or two. Spread mayonnaise mixture on both sides of the bun, place the burger on the bottom half of the bun, top with about 1/4 cup of shredded pork and place the other side of the bun on top.