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Wash and pat dry chicken breasts. Trim off any fat, pound lightly (for a more even piece of chicken).
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Combine 1/4 tsp pepper, salt, and the thyme and season chicken on both sides.
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Heat skillet on medium high, add butter and olive oil.
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Add chicken breasts and brown for 3 minutes on each side.
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Remove chicken to a platter, and add mushrooms and green onions to skillet.
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Saute mushrooms and onions, then add the wine, tarragon and remaining salt and pepper.
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Stir to combine, and then add chicken breasts back to the pan. Simmer for 20 minutes.
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Remove chicken to serving platter, and stir in sour cream.
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Heat sauce through then pour all over chicken breast. Looks pretty garnished with some fresh chopped parsley!