Preheat oven to 300°F (150°C). Pat the pork shoulder dry with paper towels. In a small bowl combine kosher salt, black pepper, smoked paprika, ground cumin, garlic powder, and onion powder. Rub the pork all over with olive oil, then evenly coat with the spice mixture.

Sear (optional)

Heat a heavy oven-safe skillet or Dutch oven over medium-high heat. Sear the pork 2–3 minutes per side until browned to develop flavor. This step is optional but recommended.

Roast slowly

Add the chicken broth, apple cider vinegar, and brown sugar to the pot around the pork (not directly over the seasoned surface). Cover tightly with a lid or foil and transfer to the preheated oven. Roast for 3 to 3.5 hours, or until the internal temperature reaches 195–205°F (90–96°C) and the meat is very tender and pulls apart easily.

Rest and shred

Remove pork from oven and let rest, covered, for 15–20 minutes. Transfer to a cutting board and using two forks or meat claws, shred the pork into bite-size pieces. Skim any excess fat from the pan juices and stir a few tablespoons of the cooking liquid into the shredded pork to keep it moist. Taste and adjust seasoning with salt and pepper if needed.

Sun-Dried Tomato Pesto Aioli

Blend aioli

Place the drained sun-dried tomatoes, mayonnaise, fresh basil, grated Parmesan, garlic clove, lemon juice, red pepper flakes, and salt into a food processor or high-speed blender. Pulse while streaming in the extra-virgin olive oil until the mixture is smooth and spreadable. Scrape down the sides and taste; adjust salt, lemon, or red pepper flakes to preference. Transfer to a bowl and refrigerate until assembly.

Coleslaw

Make dressing

In a medium bowl whisk together Greek yogurt, mayonnaise, lime juice, honey, salt, and black pepper until smooth.

Toss slaw

Add the shredded green and red cabbage, grated carrot, sliced red onion, and chopped cilantro to a large bowl. Pour the dressing over the vegetables and toss until evenly coated. Taste and adjust seasoning with additional lime juice or salt if desired. Refrigerate for at least 15 minutes to allow flavors to meld.

Taco Assembly

Warm tortillas

Warm tortillas just before serving. Heat a large skillet over medium-high heat and add a thin film of vegetable oil. Warm each tortilla 20–30 seconds per side until pliable and slightly toasted. Alternatively, wrap in foil and warm in a 300°F oven for 10 minutes.

Assemble tacos

Spread about 1 teaspoon of the sun-dried tomato pesto aioli onto each warm tortilla. Place 2–3 tablespoons of shredded pulled pork on top, then add 1–2 tablespoons of coleslaw. Garnish with fresh cilantro leaves and pickled jalapeños if using. Serve with lime wedges on the side for squeezing over the tacos.

Make ahead and storage

Pulled pork and aioli can be made up to 3 days ahead; refrigerate separately in airtight containers. Refrigerate coleslaw up to 1 day. Reheat pork gently in a covered pan with a splash of cooking liquid or in a 300°F oven until warmed through. Do not freeze the aioli.