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1
Preheat the oven to 300.
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2
Season the chicken with salt and pepper.
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3
In a large, enameled cast-iron casserole, heat 2 tablespoons of the canola oil.
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4
Add half of the chicken and cook over moderately high heat, turning, until browned, 8 minutes.
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5
Transfer to a plate.
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6
Cook the remaining chicken, then pour off the fat and wipe out the casserole.
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7
Heat the remaining 2 tablespoons of canola oil in the casserole.
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8
Add the onion, carrot, celery and shallots and cook over moderate heat, stirring, until the vegetables are softened and lightly browned, 8 minutes.
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9
Add the wine and simmer for 1 minute, scraping up the browned bits from the pot.
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10
Add the chicken stock and thyme and bring to a boil.
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11
Nestle the chicken in the casserole; cover and braise in the oven for 1 hour, until tender.
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12
Meanwhile, in a very large skillet, melt the butter in the olive oil.
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13
Add the mushrooms and cook over high heat, without stirring, until well browned, 5 minutes.
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14
Season the mushrooms with salt and pepper and cook, stirring, until tender, 3 to 5 minutes; transfer to a plate.
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15
Transfer the chicken to a plate.
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16
Strain the braising liquid through a fine sieve into a heatproof bowl, pressing on the solids; skim off the fat.
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17
Return the braising liquid to the casserole and boil until reduced to 1 1/2 cups, 3 to 5 minutes.
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18
Whisk in the creme fraiche and lemon juice and season with salt and pepper.
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19
Add the mushrooms and chicken to the sauce and simmer for 3 minutes.
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20
Garnish with tarragon and serve.
