How to Make It
Preheat the oven to 300°. Season the chicken with salt and pepper. In a large, enameled cast-iron casserole, heat 2 tablespoons of the canola oil. Add half of the chicken and cook over moderately high heat, turning, until browned, 8 minutes. Transfer to a plate. Cook the remaining chicken, then pour off the fat and wipe out the casserole.
Heat the remaining 2 tablespoons of canola oil in the casserole. Add the onion, carrot, celery and shallots and cook over moderate heat, stirring, until the vegetables are softened and lightly browned, 8 minutes. Add the wine and simmer for 1 minute, scraping up the browned bits from the pot. Add the chicken stock and thyme and bring to a boil.
Nestle the chicken in the casserole; cover and braise in the oven for 1 hour, until tender.
Meanwhile, in a very large skillet, melt the butter in the olive oil. Add the mushrooms and cook over high heat, without stirring, until well browned, 5 minutes. Season the mushrooms with salt and pepper and cook, stirring, until tender, 3 to 5 minutes; transfer to a plate.
Transfer the chicken to a plate. Strain the braising liquid through a fine sieve into a heatproof bowl, pressing on the solids; skim off the fat. Return the braising liquid to the casserole and boil until reduced to 1 1/2 cups, 3 to 5 minutes. Whisk in the crème fraîche and lemon juice and season with salt and pepper. Add the mushrooms and chicken to the sauce and simmer for 3 minutes. Garnish with tarragon and serve.