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Preheat oven to 375°F.
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Season chicken generously with salt, cayenne and herbs de Provence.
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Place in an oven safe skillet, interspersed with garlic, red pepper and carrot.
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Tuck in the parsley sprigs and pour on stock to almost cover chicken.
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Cover the skillet and braise in the oven until chicken is just cooked through.
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Meanwhile, reduce the vinegar in another saucepan to 2 Tablespoons.
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Remove the chicken from the skillet and set aside.
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Place the skillet on the stove and simmer until vegetables are very soft.
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Remove the parsley and puree the skillet's contents in a blender. (I used an immersion blender).
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Return sauce to skillet.
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Add the reduced vinegar and tomato paste to the sauce.
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Return the chicken to the pan to reheat gently.
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Serve on a platter with chopped parsley, if desired.