Start by placing the chicken pieces in a large pot. Cover them with water and bring to a boil. Once boiling, reduce the heat and let it simmer for about 30 minutes, until the chicken is cooked through. Remove the chicken from the pot and set it aside to cool.
In the same pot, melt 2 tablespoons of butter over medium heat. Add the chopped onion, celery, and carrots. Sauté until the vegetables are softened, roughly 5-7 minutes.
Sprinkle the flour over the sautéed vegetables and stir for another 2 minutes. This will help thicken your bisque. Gradually add the reserved chicken broth, stirring constantly to avoid lumps.
While the broth is coming to a simmer, shred the cooled chicken, discarding the skin and bones. Add the shredded chicken back into the pot.
Stir in the chopped pimientos, green pepper, salt, and freshly ground pepper. Let the mixture simmer for about 15 minutes, allowing the flavors to meld beautifully.
Finally, stir in the heavy cream. Simmer for an additional 5 minutes until heated through. Taste and adjust seasoning as necessary.