Combine the dry rub: In a small mixing bowl, stir together the brown sugar, smoked paprika (if using), garlic powder, onion powder, salt, and black pepper until evenly combined. Set aside.
1 tablespoon brown sugar, 1 teaspoon smoked paprika, 1 teaspoon garlic powder, 1 teaspoon onion powder, 2 teaspoons salt, 1 teaspoon black pepper
Prepare and season the pork. Trim away any thick, excessive layers of fat from the pork shoulder, leaving some fat for flavor and moisture. Pat the meat dry with paper towels. Sprinkle the prepared seasoning mixture over the entire roast, rubbing it into the surface and crevices.
4-5 pounds pork shoulder (pork butt)
Rest the pork shoulder. Allow the seasoned pork to stand for 30 minutes before searing. Preheat the oven to 325°F (163°C) toward the end of the resting time.
Brown the pork shoulder. Place a large Dutch oven over medium-high heat and add the olive oil. Once the oil is hot, carefully lower the pork shoulder into the pot. Sear for approximately 3–4 minutes per side, turning as needed to brown the exposed surfaces. Watch closely to prevent the brown sugar in the seasoning from burning. Transfer the browned pork to a clean plate, leaving the cooking juices and browned bits in the pot.
2 tablespoons olive oil
Prepare the braising liquid. Reduce the heat to medium and add the sliced red onion to the Dutch oven. Cook for approximately 4–5 minutes, stirring occasionally, until the onion begins to soften. Stir in the minced garlic and cook for another 30–60 seconds, just until fragrant. Slowly add the beer and apple cider, scraping the bottom of the pot with a wooden spoon to release the browned bits. Bring the liquid to a gentle simmer.
1 large red onion, 4 cloves garlic, 12 fluid ounces (1½ cups) beer, 12 fluid ounces (1½ cups) nonalcoholic apple cider
Return the pork to the Dutch oven. Place the seared pork shoulder back into the Dutch oven, along with any juices that accumulated on the plate. Arrange the fresh thyme and rosemary sprigs around the meat, tucking them into the braising liquid. The liquid should reach partway up the sides of the pork without completely submerging it.
2 sprigs fresh thyme, 2 sprigs fresh rosemary
Cover and braise. Cover the Dutch oven with a tight-fitting lid and transfer it to the preheated oven. Cook for approximately 3½–4 hours, spooning some of the braising liquid over the exposed pork once or twice during cooking. Keep the Dutch oven covered between bastings. Begin checking for tenderness near the end of the cooking time. The pork should pull apart easily with two forks. If it remains firm or resistant, cover and continue cooking, checking every 20–30 minutes until tender.
Rest the pork. Carefully transfer the cooked pork shoulder to a cutting board or large platter. Allow it to rest for 10–15 minutes while you prepare the sauce.
Strain the cooking liquid. While the pork rests, remove the herb stems from the Dutch oven. Pour the remaining cooking liquid and onions through a fine-mesh strainer set over a saucepan. Set the softened onions aside to add back to the sauce. Discard the herb stems and any unwanted solids.
Reduce and season the sauce. Skim off excess fat from the strained cooking liquid, if necessary. Place the saucepan over medium heat and bring the liquid to a gentle simmer. Cook uncovered until the sauce has reduced to your preferred concentration, stirring occasionally. Return the reserved softened onions to the saucepan and stir to combine. Taste and adjust the seasoning as needed.
Shred and serve. Use two forks to pull the rested pork into large pieces, discarding any bones, cartilage, and excess fat. Arrange on a serving platter or individual plates and spoon the warm braising sauce and softened onions generously over the meat. Serve immediately.