Preheat the cooker. Set the smoker or pellet grill to 375°F. For a covered grill, use indirect heat at 375°F. For the oven method, preheat to 375°F and use the center rack. Lightly grease a 9-by-13-inch metal baking pan with 1 teaspoon oil.
Carve the peppers. Wash and dry the peppers. Cut off the tops in one piece, keeping the stems attached, and reserve the lids. Remove seeds and white membranes. Place each pepper securely on a cutting board and use a paring knife to cut small triangle eyes and a simple mouth, keeping your supporting hand out of the blade’s path. Leave wide strips of pepper between cuts. Keep the base intact; use a small foil ring to support any wobbly pepper.
Sauté the onion. Heat 1 tablespoon oil in a large skillet over medium heat. Add the diced yellow onion and cook for 4–5 minutes, stirring, until softened. Clean pepper scraps can go in with the onion.
Heat the filling. Stir in the cooked pulled pork, cooked rice, BBQ sauce, Worcestershire sauce, apple cider vinegar, smoked paprika, garlic powder, and black pepper. Cook, stirring frequently, until hot throughout, about 5 minutes or longer if very cold. When using refrigerated leftover pork or rice, heat the mixture to 165°F. Add the optional water 1 tablespoon at a time if needed; the filling should be moist, not soupy. Remove from the heat and fold in 1 cup cheddar.
Stuff the peppers. Stand the peppers in the pan. Divide the hot filling among them, spooning it in gently to just below the rim. Do not pack tightly or push filling through the faces. Put any extra filling in a small cooker-safe dish to heat alongside. Place the pepper lids beside the peppers.
Cook and top with cheese. Smoke or grill over indirect heat, uncovered with the cooker lid closed, for about 25–35 minutes. Alternatively, bake in the oven for about 30–35 minutes. Add the remaining 1/2 cup cheddar during the final 5–10 minutes. Start checking the smoked peppers at 25 minutes. Continue until the pepper walls are fork-tender but still upright and the center of the filling in each pepper reaches 165°F. If the tops brown too quickly, loosely tent with foil and continue cooking.
Rest and serve. Rest for 5 minutes. Set or lean a reserved lid on each pepper and lift carefully from underneath with a wide spatula. Serve one large stuffed pepper per person.