Place the chicken pieces in a large pot and add water to cover (about 3 quarts). Cook on low heat for 1 to 1 1/2 hours or until the chicken is done and tender. Skim off any foam or fat that rises to the surface during the initial boiling.
Remove the chicken from the pot and let it cool. (Reserve the chicken stock in the pot.) Once it's cool enough to handle, shred the meat, discarding the skin and bones.
To make a roux, melt the butter in a 3-quart saucepan over low heat. Add the chopped onion and sauté until translucent. Add the flour and cook, stirring, for 1 or 2 minutes.
Bring 8 cups of the reserved chicken stock to a slow boil and gradually stir it into the roux. Simmer until the soup takes on a glaze, about 15 minutes.
Add the shredded chicken, pimientos, green pepper, ground pepper, and heavy cream. Simmer for 10 to 15 minutes, stirring occasionally. Taste and adjust the seasoning if necessary. Serve in heated bowls.
Recipe Tips
Add more cream for a creamier bisque.You can use any remaining chicken in salads or other recipes.