Pat the Boneless Pork Shoulder dry and season lightly with Kosher Salt and Black Pepper. Heat 2 tablespoons of Vegetable Oil in a large heavy pot over medium-high heat and brown the pork cubes in batches until nicely seared; transfer to a plate.
Reduce heat to medium, add the diced Onion and cook in the same pot until softened, about 5 minutes, then stir in the minced Garlic and cook 30 seconds more.
Sprinkle in the All-Purpose Flour and stir for 1 minute to form a light roux; return the seared pork to the pot and stir to coat.
Add the chopped Roasted Green Chiles, diced Tomatoes, Chicken Broth, Ground Cumin, Dried Oregano, and a little extra Kosher Salt and Black Pepper to taste. Bring to a simmer, cover, and cook gently until the pork is very tender and the sauce has thickened, about 1 to 1½ hours (or simmer longer on low for better flavor).
While the pork simmers, heat a skillet over medium heat and brown the Ground Beef with a splash of Vegetable Oil if needed. Add the diced Onion and minced Garlic and cook until the beef is no longer pink.
Season the beef with Ground Cumin, Chili Powder, Kosher Salt and Black Pepper. Cook a few minutes more, then set aside.
Warm the Canned Refried Beans in a small saucepan or microwave until smooth and spreadable; keep warm.
To assemble, warm the Flour Tortillas in a dry skillet or wrapped in foil in a low oven so they are pliable. Spread a layer of refried beans down the center of each tortilla, top with a scoop of the seasoned beef and several spoonfuls of shredded pork and green chile sauce.
Fold the sides over and roll into burritos, then place them seam-side down in a baking dish. Spoon additional green chile sauce from the pork pot over the burritos and sprinkle generously with Shredded Cheese.
Bake at 375°F (190°C) for 10–15 minutes, until the cheese is melted and the edges are heated through. Garnish with shredded Iceberg Lettuce, diced tomato, chopped Cilantro, and a dollop of Sour Cream before serving.
Taste and adjust seasoning with extra Kosher Salt or Black Pepper as needed, and serve hot with any remaining green chile sauce on the side for spooning over individual burritos.
