Preheat the oven to 350°F.
Heat a large oven-safe braising pot over medium heat with 1 tablespoon oil. Pat the pork dry and season with salt and pepper. When the oil is hot, add the pork, working in batches to avoid crowding the pan, and sear on all sides. Add more oil as necessary to sear. Transfer to a plate.
Add the onions and garlic to the po and cook until soft, 5-7 minutes. Add the spices and toast for 1 minute. Add the green chiles and chicken broth and bring to a simmer for 20 minutes. Purée the salsa verde with an immersion or standing blender until smooth. Adjust the seasoning to taste.
Return the seared pork to the dish with the salsa verde. Cover with an inverted piece of parchment paper touching the top of the beef and a tightly-fitting lid. Braise for 90 minutes to 2 hours, until the meat is falling-apart tender.
Shred the beef with two forks, adding braising liquid as necessary to moisten.
On a baking sheet, toss the roma tomatoes, jalapeños, and yellow onion with the canola oil. Roast for 10-15 minutes, until browned. Transfer to a blender, along with the remaining roasted salsa ingredients. Blend until smooth and season to taste.
In a large bowl, combine the salt, sugar, vinegar and coriander seeds until the salt and sugar are dissolved. Fold in onion and cabbage.
Lay the tortillas on a work surface. Fill with shredded pork and both cheeses. Fold to close.
Heat a large skillet with 1 tablespoon canola oil over medium-high heat. Cook each quesadilla for about 2 minutes per side, until golden and the cheese is melted. Serve with pickled cabbage, roasted salsa, sour cream and a lime wedge.