Cut the pork shoulder into large chunks and season generously with salt and pepper.

Heat olive oil or lard in a large Dutch oven over medium-high heat. Sear the pork until browned on all sides, about 10 minutes.

Remove the pork and set it aside to let the juices settle.

In the same pot, add the chopped onion and sauté until translucent, about 5 minutes. Then, add minced garlic and sauté for an additional minute.

Add halved tomatillos, roasted Hatch green chiles, ground cumin, and Mexican oregano to the pot. Cook until fragrant, roughly 3 minutes.

Return the seared pork to the pot and pour in 2 cups of chicken broth. Bring to a boil, then reduce the heat and simmer for 2 to 3 hours.

Once the pork is tender, carefully remove it from the pot and shred it.

Allow the sauce to simmer uncovered for about 15 minutes to reduce.

Stir chopped cilantro into the sauce, then add the shredded pork back into the pot.

Lay out large flour tortillas, spoon the pork mixture and cheese onto each one, then roll them into burritos.

Heat a skillet over medium heat and toast the burritos until golden brown on both sides, roughly 3-5 minutes.

Plate the burritos, smother with extra verde sauce and melted cheese, and garnish with fresh cilantro.