Cook, stirring frequently, until the chicken is cooked through, about 8 minutes. Add the chicken slices and brown slightly. Stir the flour into the chicken mixture. Add the beans, green chiles, lime juice, and chicken stock. You can mash some of the beans to thicken the chili. Bring the mixture to a simmer, scraping up the brown bits that cling to the bottom of the pan with a wooden spoon.
Simmer for 55-60 minutes until the liquid has reduced and the chili has thickened. If you like the chili even thicker, just mix a couple of tablespoons of cornstarch with about a quarter cup of cold water and pour in the pot. Add the Parmesan cheese and cilantro and simmer for another 10 minutes. Season with salt and pepper, to taste.
Ladle the chili into serving bowls. Drizzle with salsa verde and top with sour cream.