Pat the pork shoulder dry using paper towels. Sprinkle liberally with salt on all sides. Set aside.
In a large bowl whisk to combine the citrus juices, vinegar, garlic, herbs, cumin, and pepper. Lower the pork into the marinade and cover. Place the bowl in the fridge and let rest for at least 8 hours or overnight.
Preheat your oven to 275°F. Or soak 1-2 cups of good quality wood chips. Build 2 beds of hot coals on either side of your charcoal BBQ and sprinkle the soaked wood chips on top. Cover and adjust the vents of your BBQ to coax it up to 275° F.
When your oven or BBQ is primed, cover the base of a dutch oven with the onion wedges. Place the pork on top and pour in half of the marinade. Cover and place in the oven or on the grill.
Cook the pork for 2 hours covered, then remove the cover and cook for at least 2 hours more. If you want to be able to slice the pork, cook it to an internal temperature of 170°F. If you want it shreddable, like in the photos above, cook the pork to 190°F.
Take the pork off the heat and tent in foil. Let rest for 30 minutes. Pour off the juices from the pot and place in them in the freezer.
Once the pork has rested, take the reserved pork juices out of the freezer and scrape off the fat that will have settled on top. Discard. Spoon the rest into a small skillet and bring to a boil, then reduce to a simmer. Cook until reduced slightly, about 6-7 minutes.
While the sauce is reducing, shred the pork and remove any and all bones. Pour the reduced pork juices over the meat and toss to coat. Cover and keep warm while you create the rest of the burger.