Preheat the oven to 350 F.
Season the chicken with the salt and pepper. Warm 1 teaspoon of the oil in a large nonstick saute pan over medium heat. Carefully, lay the chicken breasts, skin side down, in the pan. Sear for 3 minutes, or until the skin has nicely browned. Transfer the chicken, skin side up, to a nonstick jelly-roll pan. Reserve the saute pan.
Bake the chicken for 20 minutes, or until the juices run clear when the chicken is pierced with a knife.
While the chicken is cooking, remove most of the fat from the saute pan. Add the carrots and onions to the pan. Saute over medium heat for 3 minutes, or until the vegetables begin to caramelize. Add the stock and raise the heat to high. Bring to a boil, then reduce the heat to medium. Simmer for 10 minutes, or until the liquid has reduced by half. Strain through a fine sieve into a small bowl; skim off any fat that rises to the surface.
Warm the remaining 1 teaspoon oil in a medium saute pan over medium heat. Add the mushrooms and shallots. Saute for 5 minutes, or until the mushrooms are golden. Season with the salt. Remove the pan from the heat and add the cognac. Carefully ignite the cognac with a long match. Allow the flame to burn out, then add the wine. Return the pan to the heat.
Bring to a boil over medium-high heat. Reduce the heat to medium and simmer for 10 minutes, or until the liquid has reduced by half. Add the reserved stock and simmer for 5 minutes, or until the sauce is thick enough to coat the back of a spoon. Stir in the tarragon. Taste and adjust the seasoning.
Remove the skin from the chicken breasts. Carefully cut the breast meat from the breast bone and then cut each breast, on the diagonal, into 2 pieces. Place a divided chicken in the center of each of 4 warm dinner plates.
Recipe courtesy of The French Culinary Institute's Salute to Healthy Cooking by Alain Sailhac, Jacques Pepin, Andre Soltner, International Culinary Center, Jacques Torres