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Roast the Anaheim, poblano, and jalapeños chiles under the broiler until blackened, about 5 minutes per side. Place the chiles in a paper sack or plastic food-storage bag, close it tight and let the chiles steam for 20 minutes. When done, take the chiles out of the bag and gently rub off the skin of each chile. Remove the stems and seeds, and then dice.
Meanwhile, sprinkle the beef with the salt and pepper. Heat the bacon grease or oil on medium-low in a large heavy pot, such as a Dutch oven, and then working in batches brown the beef on all sides. This should take about 5-7 minutes per batch. If using a slow cooker, after the beef has browned, remove it from the pot and place in the slow cooker. If using the Dutch oven, place the browned beef into a large mixing bowl.
Once the beef has been browned, leaving the heat on medium-low, add the onions to the pot and cook until translucent, about 5 minutes. Add the garlic and cook for another 30 seconds.
If using the large pot to cook the beef green chili, pour into the pot the chicken broth, and then scrap the bottom of the pot to incorporate all the cooked bits into the broth. Add the chopped chiles, the cooked beef, the tomatillos, half of the chopped cilantro, cumin, oregano, and allspice to the pot.
Bring to a boil and then turn the heat down to low and simmer uncovered for 3 hours or until the beef is fork tender. At this time, add the remaining cilantro and then taste and adjust seasonings. Mix the masa harina with 1/4 cup of water and then stir into the pot until well combined. Continue to cook for 30 more minutes and then add the lime juice. Serve warm with sour cream, cilantro, warm tortillas and/or tortilla chips.
If using a slow cooker, transfer the onions and garlic from the pot into the slow cooker. Leaving on the heat, pour into the Dutch oven the chicken broth, and then scrape the bottom of the pot to incorporate all the cooked bits into the broth. Pour the broth into the slow cooker and add the chopped chiles, the tomatillos, half of the chopped cilantro, cumin, oregano, and allspice. Cover the slow cooker and cook on high for 4 hours or low for 6.
After it’s cooked on high for 4 hours or low for 6, remove the lid from the slow cooker and stir in the remaining cilantro. Taste and adjust seasonings. Mix the masa harina with 2 tablespoons of water and then stir into the pot until well combined. Continue to cook 30 more minutes uncovered and then stir in the lime juice. Serve warm with sour cream, cilantro, warm tortillas and/or tortilla chips.