Move the oven rack to the lower-middle position and preheat the oven to 300 degrees F. Trim the beef and cut it into 1 ½-inch pieces. Use paper towels to pat the beef dry and then season the meat on all sides with salt and pepper. Add the oil to a large heavy-bottomed pot and heat over medium-high heat until the oil is shimmering and just beginning to smoke. Add in the meat and cook until the meat is browned about 8-10 minutes.
Take the pot off of the heat and stir in the enchilada sauce. Put the lid on the pot and cook in the preheated oven for 2-2 ½ hours or until the meat is fork-tender.
Set a large strainer over a large bowl and empty the beef and sauce into the strainer. Move the meat to a large bowl or pan and use two forks to shred the beef. Reserve the strained off enchilada sauce for later in the recipe (there should be about 2 cups of sauce).
Turn the oven temperature up to 375 degrees F. Spoon ⅔ cup of the reserved enchilada sauce into the bottom of a 13x9-inch baking pan. Add 1/3 cup of the enchilada sauce and 1 cup of the shredded cheese to the beef. Stir until the beef is coated in the sauce and the cheese is evenly distributed.
Heat all of the tortillas in the microwave until they are warm and pliable, about 30-60 seconds. Place a tortilla on a plate or countertop and add 1/3 cup of the beef filling into the middle. Roll up the tortilla nice and tight and place it seam-side down into the 9x13 pan. Repeat with the remaining tortillas and filling.
Use a spoon to evenly distribute the remaining enchilada sauce over the enchiladas. Then, evenly distribute the remaining shredded cheese and then cover the enchiladas with foil. Bake for 22 minutes, or until the enchiladas are hot and the cheese is melted. Serve the enchiladas on plates and garnish with sour cream, cilantro, avocado, and lime wedges.