Grind 1/2 cup of the walnuts in a food processor or blender and set aside.

Rinse the chicken quickly under cold water and dry with paper towels. In a large skillet, heat 1-1/2 tablespoons of the butter and the oil over moderately high heat and in it brown the chicken pieces in batches, setting them aside as they are done.

Pour off all but 2 tablespoons of fat. In the skillet, over moderately low heat sauté the shallots until they are lightly browned. Set aside.

Add the wine to the skillet and cook over high heat for a few minutes, scraping up the brown bits at the bottom of the skillet. Stir in the ground walnuts and return the chicken and any accumulated juices to the skillet, placing the dark meat on the bottom. Add the shallots, pushing them down into the wine/walnut mixture. Add salt and pepper to taste, sprinkle with the tarragon, and pour in the stock.

Bring the mixture to a simmer, cover, and cook slowly until the chicken is tender, about 30 minutes.

While the chicken is cooking, heat the remaining 1/2 tbsp butter over moderately high heat in a small skillet and in it sauté the remaining walnut pieces for a few minutes, stirring constantly so they do not burn. Set aside.

When the chicken is tender, remove it and the shallots to a warmed serving dish.

Strain the sauce through a sieve into a small saucepan and reduce it over high heat until a little more than 1 cup remains. Add the cream, lower the heat, and boil gently until the sauce has thickened, 2 or 3 minutes. Pour the sauce over the chicken and top with the walnuts.