Place the pork shoulder in the slow cooker, then pour in the broth and spread barbecue sauce over the top. Cover tightly with the lid and cook on low for 4 hours until the pork becomes fall-apart tender.
Once cooked, use tongs to carefully remove the pork from the slow cooker and transfer it to a large plate or cutting board. Let it rest for a few minutes, then shred the pork with two forks until it’s pulled into bite-sized pieces, releasing a smoky aroma.
Meanwhile, heat a skillet over medium heat. Add sliced onions and peppers, sautéing until they soften and start to caramelize, about 5-7 minutes. They should be fragrant and tender with a slight char.
In a small bowl, mix cumin, smoked paprika, minced garlic, and a squeeze of lime juice to create a quick seasoning blend. Toss the shredded pork with this mixture to boost the smoky flavor and add brightness.
Heat a clean skillet or griddle over medium-high heat. Place a flour tortilla on the hot surface and spread a handful of shredded pork over half of it. Top with sautéed peppers, onions, and shredded cheese.
Fold the tortilla in half over the fillings, pressing gently. Cook for 2-3 minutes until the bottom is golden brown and crispy, then flip and cook the other side until equally crispy and the cheese is melted.
Remove the quesadilla from the skillet and let it rest for a minute. Slice into wedges and serve hot, with extra lime wedges if desired.
