Pull off the husks of the tomatillos and give them a good rinse. I usually de-stem them but this is optional. Add the tomatillos to a roasting pan along with the rinsed poblanos and roast them in the oven at 400F. I usually flip the poblanos over after 15-20 minutes, and if the tomatillos are disintegrating you can take them out of the oven at this point. The poblanos will need about 30 minutes total to fully roast.
Add the two chicken breasts to a pan and cover with 2 quarts of stock. Be sure there is enough liquid to submerge the chicken breasts. I also added 1/2 onion, 10 sprigs cilantro, a pinch of salt, and some freshly cracked black pepper but these are optional. Bring to a boil and then reduce heat to a simmer. Cook the chicken until no longer pink inside, about 20 minutes. Once cooked, set the chicken aside to cool on a plate and then shred using two forks. Save the cooking liquid.
Add the remaining green sauce ingredients to a blender: 2 peeled onions, 2 de-stemmed and rinsed jalapenos, 4 peeled garlic cloves, and 1/2 bunch of rinsed cilantro. I usually just twist off and discard the bottom, thicker portions of the cilantro but use the upper stems that hold the leaves together.
Once the poblanos have fully roasted, remove them from the oven and let them cool down for a few minutes. Pull off and discard as much of the skin as you can. De-stem and de-seed them. I find it easiest to cut off the tops and then make a slit lengthwise. Open up the pepper and use the knife to scrape out the veins and seeds.
Add the poblanos and tomatillos to the blender and combine well with the other ingredients. You might have to blend half first to get all of it to fit into a single blender.
Add a dollop of oil to saucepan or Dutch oven over medium heat. Add the green sauce from the blender and cook for a few minutes.
Drain and rinse 2 cans of hominy. If you want a soupier version then use only a single can. Add the hominy to the green sauce.
Add the shredded chicken to the pan along with the chicken cooking liquid (I gave the cooking liquid a quick strain to remove the onion and cilantro bits). We'll also add 1 teaspoon Mexican oregano, 1 teaspoon salt, and some freshly cracked black pepper. You can always add additional liquid at this point if you want a thinner version (stock or water).
Let simmer for 15 minutes or until the hominy is heated through. Take a final taste for seasoning, adding more salt if necessary.
Serve immediately with a squeeze of lime. You can optionally serve it with a cheese quesadilla. Simply add cheese to half of a flour tortilla and fold onto itself. Cook in a dry skillet over medium-high heat until both sides are crispy and golden brown.
Store leftovers in an airtight container in the fridge.