Heat 2 tbsp oil over medium heat in a large soup pot. Add in the onion, garlic, poblanos and stir. Cook, stirring occasionally until translucent. Add the cut up chicken and continue cooking, stirring once in a while until it is browned a bit.
Add in all the spices and the beer. Bring to a boil. Reduce heat to a simmer and add in the chicken stock, white beans, 1/2 the hominy and the can of green chilies. Simmer for 30 minutes, tasting and adjusting the salt and seasoning. While it is cooking, make the hominy.
Dry the hominy with paper towels. Heat 1/4 inch of oil in a skillet over medium high. Toss the hominy with the cornstarch and some salt. Add to the oil and fry, shaking the pan or stirring, until it browns up. Some of the kernels may pop so you can keep a lid or screen nearby. When the hominy is browned and crunchy, drain on paper towels and sprinkle with a bit more salt if needed.
Whisk the sour cream and cream into the chili and give it a final taste. Dish up into bowls and top with cilantro, jalapeno, cheese and hominy.