Place the green chilies and ½ a cup of chicken stock in a blender or food processor and puree until smooth. Briefly set aside until ready to use.
Heat the olive oil in a medium Dutch oven or pot. When hot, add the minced onions and garlic and sauté for 3 minutes until the onions become translucent and the garlic is fragrant. (Be careful not to burn this mixture, people!) Add the oregano, cumin and chili powder and cook for 1 more minute.
Add the chili puree, shredded chicken, hominy and remaining 3½ cups of chicken stock to the Dutch oven/pot and bring the mixture to a boil. Lower the heat to a simmer and cook for 10 minutes. Taste your posole and season with salt.
Ladle the posole into bowls and garnish with fresh lime juice and toppings of your choice.