Preheat the broiler and place the poblano chiles in a skillet or on a baking sheet. Broil the chiles for about 5 minutes per side. Immediately place the chiles in a paper sack or plastic food-storage bag, close it tightly, and let the chiles steam for 20 minutes. To peel the chiles, remove them from the bag and gently rub off the skins. Remove the seeds, dice them and set them aside.
In a large soup pot or Dutch oven, heat 1 tablespoon of the bacon grease over medium-low heat. Lightly salt and pepper the pork and add to the pot, browning on each side, 2 to 5 minutes per side. You may have to do this in batches. Once browned, transfer the pork to a large bowl, pouring the pan juices into the bowl.
Return the pot to the stove and heat the remaining 1 tablespoon of bacon grease over medium-low heat. Add the onion and cook until translucent, about 5 minutes. Add the garlic and cook for 30 seconds more.
Return the pork and its juices to the pot. Add the poblano chiles, jalapeno chiles, tomatillos, chicken broth, Mexican lager, cumin, oregano, allspice, cay enne, and 1/2 cup of the cilantro. Turn the heat to high and bring to a boil; then turn the heat down to low and gently simmer, uncovered, for 2 hours, stirring occasionally and skimming any fat from the surface.
Taste and adjust the seasonings, adding salt and pepper to taste. Continue to cook, uncovered, until the pork is tender, anywhere from 30 to 60 minutes more. To thicken the chili, in a bowl mix together the masa harina with 2 table spoons of the chili liquid. Stir this back into the pot, along with the remaining 1/2 cup cilantro and the lime juice. Cook until the chili has thickened, 10 more minutes. Serve with sour cream and shredded Monterey Jack cheese.
Adapted with permission from The Homesick Texan's Family Table by Lisa Fain, © 2014. Published by Ten Speed Press, an imprint of Random House LLC.