Preheat the oven to 325 degrees F.
To prepare the pork: combine sugar, salt, pepper, paprika, mesquite, cumin, and oregano in a bowl. Optional to trim fat from pork ( I did not). Rub the pork on both sides with mustard using your hands to cover most of the shoulder. Season with the dried spice mixture all over ensuring fully covered. Place pork in a large oven safe dish and toss in those 4 garlic cloves. Pour the orange juice into the dish from the side ensuring not to remove any of the rub or spice from the meat. Do the same with the broth. Cover with foil and Cook for 3 1/2 hours until tender. Depending on thickness of the shoulder it could take up to 4 hours. If the meat isn't pulling off easily with hands, return to oven.
Remove from the oven and shred the meat. Keep the shredded meat in its juices and discard fat here. The fat should easily come apart and be noticeably visible.
Increase the oven temperature to 400 degrees F.
Gently run a bread knife along the side of the rolls going across to cut open the top. Use your hand to hold down the tops and keep it steady. Mix the same yellow mustard, about 1/3 cup with 1/4 cup of the braising liquid from the pork. You should have a mustard that's easily spreadable. Spread acrosss the bottom bun to fully cover, Top with a good amount of shredded pork, next layer the ham and the cheese slices. Place 2 buttered pickles on top of the cheese per slider, close the sliders with the top buns and cover with a piece of foil. Bake for 20 minutes until warmed through and the cheese is melted.