Preheat the oven to 375 degrees F.
Remove the wings from the turkey and set aside on a baking sheet or large plate. Remove the legs and then separate the thighs from the drumsticks; place on the baking sheet. With kitchen scissors, cut along the ribs to remove the backbone from the turkey, leaving behind a full bone-in breast. (Alternatively, ask your butcher to cut the turkey into parts.)
Sprinkle the wings, thighs and legs with salt and pepper on both sides. In a large braiser or Dutch oven, heat the olive oil over medium-high heat. Working in batches, brown the wings, legs and thighs until golden and crisp, about 5 minutes per side. Remove from the pan and set aside.
Add to the pan 4 tablespoons of the butter, the celery, leeks and onions, and 2 tablespoons of the chopped herbs. Cook, stirring, until the vegetables have softened, 5 minutes. Add the flour, stirring until completely incorporated. Pour in the cognac and 4 cups of the chicken stock and bring to a simmer over medium heat. Return the seared turkey parts to the pan, skin-side up, without submerging them completely in the liquid. Reduce the heat to medium low, cover and cook until the meat is tender and easily pulls away from the bone, about 2 hours.
Meanwhile, rub the remaining 4 tablespoons butter over the skin of the turkey breast. Sprinkle with the remaining 2 tablespoons chopped herbs and some salt and pepper. Place the breast on a rack over a roasting pan and transfer to the oven.
Roast for 45 minutes. Add the remaining 4 cups chicken broth to the roasting pan and baste the turkey with the pan juices. Continue to roast, basting every 30 minutes and adding more chicken broth to the pan as needed, until the internal temperature reaches 160 degrees F, another 1 1/2 to 2 hours. Remove from the oven and transfer the breast to a cutting board. Let rest for 30 minutes, reserving pan juices.
Remove the wings, legs and thighs from the braising liquid and arrange them on a serving platter. Carve the breast and arrange it on the platter alongside the turkey parts. Add the reserved turkey breast pan juices to the braising liquid in the braiser and bring to a simmer. Season with salt and pepper, then strain the gravy and serve.