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Generously salt and pepper meat on all sides.
In a large dutch oven (or skillet if you’ll be using a slow cooker), heat oil over medium-high heat. Brown meat on all sides, in batches if necessary.
Add onion and just enough chicken or beef broth to mostly submerge the meat. Stir, scraping the bottom of the pan to remove any stuck-on bits. Stir in lime juice, apple cider vinegar, optional minced chipotle peppers, cumin, oregano, optional ground cloves, salt, and bay leaves and bring mixture to a low simmer. Cover and cook, stirring occasionally for 2 1/2 to 3 hours, until meat is very tender and can be pulled apart with a fork.
If using a slow cooker, transfer meat and any drippings remaining in the pan to the slow cooker and add onions, 1 cup of broth and lime juice, apple cider vinegar, optional minced chipotle peppers, cumin, oregano, optional ground cloves, salt, and bay leaves. Cook on high for 3 to 4 hours, low for 7 to 8 hours, until the beef is tender and can be easily shredded with a fork.
Once meat is done, use a slotted spoon to transfer the pieces to a cutting board. Use two forks to shred the meat, discarding any large pieces of fat as you go.
Add more salt and pepper to taste. Measure out 2 cups of the meat and set aside. If you have more than 2 cups, you can return the extra meat to its juices and keep it for other uses (or just eat it!).
Cook the rice according to package instructions, replacing the water with chicken broth and adding 1/2 teaspoon of salt (if package instructions already call for additional salt, do not add more than the package calls for).
Stir in minced cilantro and set aside to cool.
While your rice cooks, pour your drained and rinsed beans into a bowl and use a potato masher or fork to mash about half of the beans. Set aside.
In a large skillet, heat vegetable oil over medium heat and add onions. Cook for about 10 minutes, until onions are mostly softened. If they begin to brown too quickly at any point, turn the heat down.
Add tomato paste, garlic, and spices, and cook until paste just begins to caramelize and stick to the bottom of the pan and spices are fragrant, 1 to 2 minutes.
Add beans and chicken broth and stir until most of the liquid has evaporated, 2 to 4 minutes.
Add shredded beef to the pan and stir until well-mixed. Add lime juice and salt and stir to combine. Add more salt to taste if necessary. If making burritos to freeze, cool filling to room temperature, 20 to 30 minutes.
Gently heat tortillas in the microwave by wrapping them in a barely damp dish towel and microwaving for about 1 minute.
Fill each tortilla with 1/2 cup of rice, 1/2 cup of beef filling, and 1/4 cup of cheese and any other fillings you want to include. Fold into a burrito.
Add any desired toppings, serve and enjoy!