Set the oven to broil. Place whole poblano chiles on a baking sheet and broil until blackened, about 5 minutes per side. Watch closely.
Cut chicken into bite size pieces. Sprinkle with salt and 1 tsp of oregano.
Heat the olive oil in a large pot and add the chicken. Cook until browned and then remove from the pot and set aside. Turn off the heat under the pot.
Drain and rinse the hominy in a colander to remove any bitter taste.
Add the chopped onion, minced garlic and remaining 1 tsp of oregano to the pot. Cook on medium high for 4 minutes until the onions begin to soften. Add the chili powder and cook for another minute, mixing well.
Add the chicken stock, hominy and chicken to the pot. Turn the heat to high and bring to a boil uncovered. Once boiling, turn the heat down to simmer and put the lid on. Simmer for 25 minutes.
While the pozole simmers, peel the papery skins off the peppers. Remove the stems and rinse out the seeds. Cut the chiles into bite size pieces.
Add the poblanos to the pozole and cook for another minute, mixing.
Let cool, and divide into 5 bowls. Add 1 TBSP (or less depending on spice) of sour cream to each bowl and mix in. Garnish with radishes, cilantro and lime.