Preheat the oven and prep potatoes. Heat your oven to 425°F (220°C). Pierce the sweet potatoes a few times with a fork. Place them on a baking sheet lined with foil or parchment.
Roast the sweet potatoes. Bake for 45–60 minutes, depending on size, until very tender and caramelized at the edges.
You should be able to easily pierce through with a knife.
Start the pork. While potatoes roast, place sliced onion and minced garlic in the bottom of a slow cooker or Instant Pot. Add pork shoulder, broth, smoked paprika, cumin, chili powder, salt, and pepper.
Cook the pork. Slow cooker: Cook on Low for 8–9 hours or High for 4–5 hours, until fork-tender.
Instant Pot: Cook on High Pressure for 60 minutes, then natural release for 10–15 minutes.
Oven braise (backup method): Place ingredients in a Dutch oven, cover, and bake at 300°F (150°C) for 3–3.5 hours.
Shred and sauce. Transfer cooked pork to a large bowl and shred with two forks. Skim excess fat from the cooking liquid.
Toss the pork with 1/2 to 3/4 cup of the strained cooking juices, the barbecue sauce, and apple cider vinegar. Taste and adjust salt.
Make the quick slaw. Stir together Greek yogurt, Dijon, vinegar, honey, salt, and pepper. Fold in shredded cabbage until lightly coated.
Adjust seasoning to taste.
Assemble. Slice each sweet potato lengthwise and gently fluff the inside with a fork. Pile on a generous portion of pulled pork. Top with a scoop of slaw, green onions or cilantro, and jalapeños if you like heat.
Finish with a squeeze of lime.
Serve hot. Pair with a simple side salad or steamed veggies if you want extra greens.