Preheat the oven to 275 degrees Fahrenheit with a rack in the center.

Trim excess fat from the pork shoulder.

In a small bowl, combine the brown sugar, salt, onion powder, garlic powder, paprika, cumin, mustard powder, and cayenne, if using. Stir to evenly distribute.

Rub the spice blend all over the pork, making sure to get inside all the cracks.

Heat oil over medium-high heat in a Dutch oven or oven proof heavy-bottomed pot with a lid.

Add the pork shoulder to the Dutch oven. Cook, undisturbed, to lightly sear the meat, about 1 minute per side. Remove from heat.

Add the vinegar and chicken broth. Cover and transfer to the oven on the center rack. Cook until starting to soften, about 3 hours.

Remove the lid from the Dutch oven and continue cooking until the pork shoulder roast is very tender and pulls apart with no resistance, 1–2 hours (the internal temperature of the pork should be at least 200 degrees Fahrenheit). If the pork is completely dry on the bottom before it’s finished cooking, add ½ cup water or chicken broth as needed. Remove from the oven.

Using two forks, shred the pork as finely as desired directly in the Dutch oven. (Alternatively, you can transfer the pork to a cutting board to shred, if desired.)

In a medium saucepan, combine all of the barbecue sauce ingredients. Bring to a simmer over medium heat.

Cook, whisking occasionally until slightly thickened, about 10 minutes.

Remove from heat. Season with salt to taste, and add more hot sauce as desired.

Serve the shredded pork with barbecue sauce, pickles, and sandwich buns.

Pulled pork can be made 1–2 days ahead. Reheat leftover pulled pork with ¼ cup water to ensure the meat doesn't dry out.