Cut three tortillas into ninths, place in a food processor and chop until fine. Add flour, chili powder, cumin, salt and pepper. Blend to the consistence of cornmeal.
Place oil in stockpot over medium-high heat. Add onion, poblano pepper and garlic. Saute until onion is clear. Add butter and let it melt.
Add tortilla-flour mixture to the pan and mix to form a roux. Cook 4-5 minutes, stirring with a wire whisk. Do not let mixture burn. While stirring, slowly add stock, scraping down sides and bottom often. Add half and half. Bring to a slow simmer and cook 7-10 minutes. Do not let soup come to a hard boil.
Turn off heat and let cool. Add chicken before serving. Top each serving with shredded cheese and tortilla strips.