To make the poblano salsa verde, turn on the broiler and place a rack 5 inches away from heating element. Line a cast-iron skillet or baking sheet with foil and place the poblano chiles, jalapeños, tomatillos, and garlic on the skillet. Cook under the broiler for 5 minutes, and then remove the skillet from the oven. Remove the garlic from the skillet and place into a blender. Turn over the poblano chiles, jalapeños, and tomatillos, and return the skillet to the oven.
Continue to broil the chiles and tomatillos for 5 to 7 more minutes or until they are nicely charred. After this time, remove the skillet from the oven. Place the tomatillos in the blender, and put the chiles into a paper sack or plastic food-storage bag, close it tight and let the chiles steam for 20 minutes.
Pour the 1/2 cup of water into the foil-lined skillet, swirl it around, and then pour this into the blender.
After the chiles have steamed, remove from the bag and rub off the skin. Remove the stems and seeds from the chiles and add them to the blender, along with the cilantro. Blend until smooth. You should have about 3 cups of salsa. Stir in the salt, taste and add more salt if you like.
Meanwhile, to cook the pork, place it in a Dutch oven and barely cover with water. Add the garlic, lime juice, and salt, bring the pot to a boil, and then turn the heat down to low and simmer uncovered for 2-2 1/2 hours or until the meat is fork tender.
Once the meat is done, remove it from the pot and place into a wide mixing-bowl. With 2 forks, shred the meat and stir in 2 tablespoons of the cooking liquid (reserving the rest for another use, if you like) and 1/4 cup of the poblano salsa verde. Taste and add salt if needed. (Both the pork and salsa verde can be made ahead of time and kept refrigerated for 3 days.)
To make the casserole, preheat the oven to 375°F.
In a skillet, heat up the vegetable oil on low heat. One at a time, heat up the tortillas in the skilet until soft and pliant. After cooking, wrap in a cloth to keep warm. (It’s fine if you want to skip this step but note that the tortillas may get super soggy when they bake.)
To assemble the casserole, ladle 1/2 cup of the poblano salsa verde into a 9-inch square pan or a 10-inch cast-iron skillet. Place 4 of the tortillas in the pan, tearing in pieces to fill in any gaps.
Evenly top the tortillas with half the pork, 1/2 cup of shredded Monterey Jack, half the diced red onions, half the chopped cilantro, and 1/2 cup of the salsa. Top that layer with 4 more tortillas and then add the rest of the pork, 1/2 cup of the cheese, the rest of the onions, cilantro, and 1/2 cup of the salsa.
For the final layer, top with the remaining 4 tortillas. Evenly pour over them the rest of the salsa and then top evenly with the rest of the cheese.
Bake uncovered for 30 minutes or until the casserole is lightly browned and bubbling. Allow to rest for 5 minutes before serving. Serve warm topped with sour cream and cilantro.