Place the chicken breasts in a saucepan and cover with the chicken stock. Bring to a boil, reduce to a simmer, and poach for 10 to 12 minutes.
Remove the chicken and let cool, then slice thinly.
Set the stock aside to cool.
Place the tomatillos, scallions, jalapeño peppers, poblano chiles, garlic, lime juice, cilantro, and 1¼ cups of the hominy in a blender with the reserved chicken stock and pulse until very smooth. You may need to do this in batches. You can make the soup up to this point a day in advance and keep refrigerated.
When you are ready to serve, prepare your chosen garnishes.
Pour the soup into a large saucepan, add the remaining ¾ cup of hominy, gently bring to a boil, and quickly reduce the heat. It’s important not to overheat or boil it as the color will quickly diminish.
When ready, taste for seasoning and bring to the table with a ladle for people to help themselves. Serve with the garnishes and sliced chicken.