Add avocado oil to a dutch oven or braiser, and heat it over medium-high heat. Add large chunks of beef, and season well with salt and pepper. Brown all sides, and then remove from the pot. Do not overcrowd the pan. The beef will brown better and not steam if there is space around it.
Repeat with the rest of the beef, and remove all from the pan and set aside.
Add more avocado oil into the pan, and saute the onions and garlic for about 5 minutes to soften. Add all the seasonings, including the chopped up chipotle peppers, bay leaves, cumin, and Mexican oregano.
Add in the beef broth, and stir to combine.
Return the beef to the pot. Bake the beef at 300º F for 3-4 hours, until the beef falls apart when pressed with a fork.
Shred each chunk of beef with two forks. It should come apart easily. Mix the beef with the remaining broth and add the lime juice to season.
Serve the beef garnished with lime wedges and fresh cilantro, if desired.