How to Make It
In a large pot, heat 1 tablespoon of the oil over moderately high heat. Add the sliced mushrooms and 1/4 teaspoon of the salt and cook, stirring occasionally, until brown, about 5 minutes. Remove.
Reduce the heat to moderately low. Add the remaining 1 tablespoon oil to the pot. Add the onions and celery. Cook, covered, for 4 minutes. Add the garlic and cook 1 minute longer.
Add the chicken, stock, dried tarragon, if using, and the remaining 11/2 teaspoons salt. Bring to a boil. Reduce the heat and simmer for 10 minutes. Add the squash and simmer until the chicken and squash are done, 15 to 20 minutes longer. Remove the chicken from the soup. Cut the meat off the bone and then into 1-inch pieces. Skim any fat from the surface of the soup. Return the chicken to the soup. Add the mushrooms, parsley, fresh tarragon, if using, and cream, and heat through.