Heat the oven to 425F. Scrub and dry the sweet potatoes, then prick each one four or five times with a fork.
Rub the skins with olive oil and kosher salt and set them directly on a foil-lined sheet pan. Do not wrap them in foil, which steams the skin instead of drying it.
Bake 45 to 55 minutes, until a knife slides into the thickest part with no resistance.
Cook and Sauce the Chicken
Pat the chicken thighs dry and season both sides with the smoked paprika, garlic powder, salt and pepper.
Heat the neutral oil in a large skillet over medium-high heat until it shimmers. Lay in the thighs without crowding and sear undisturbed for 5 to 6 minutes.
Flip, add the water, cover, and cook 8 to 10 minutes more until the thighs reach 175F. Thighs need the extra 10 degrees over 165F to shred properly.
Rest the chicken 5 minutes on a board, then shred with two forks while it is still warm.
Off the heat, stir the barbecue sauce and 1 tablespoon cider vinegar into the pan juices, return the shredded chicken, and toss to coat. Taste against a plain bite of sweet potato and add more vinegar if it reads sweet.
Make the Slaw and Assemble
Toss the sliced cabbage with 2 tablespoons cider vinegar, the honey, olive oil and salt. Make this first so it softens for about 20 minutes.
Cut a lengthwise slit in each baked potato and press the ends toward each other to open it. Fluff the flesh with a fork.
Fill each potato with the barbecue chicken, top with the slaw and sliced green onions, and serve hot.
Notes
Thighs over breasts: thigh meat shreds into soft strands and stays juicy. If using breasts, pull at 165F and add an extra spoon of pan juices to the sauce.
Rotisserie shortcut: use about 4 cups of pulled rotisserie meat, skip to the saucing step, and add an extra pinch of smoked paprika.
Sauce off the heat: simmering chicken in barbecue sauce dulls it. Toss at the end instead.
Cut the sweetness: a sweet potato plus a sweet bottled sauce needs acid. Taste the chicken against a plain bite of potato, not on its own.
Meal prep: store potatoes whole and unsplit, chicken in its sauce, cabbage undressed. Assemble hot.
Storage: fridge 4 days, sauced chicken freezes 3 months. Reheat potatoes 15 minutes at 400F rather than in the microwave.