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Pat chicken with paper towels and sprinkle with 1 tsp salt and ½ tsp pepper. Heat butter and oil in 12 inch skillet over medium-high heat until butter is melted. Brown chicken well, about 4 minutes per side, transfer to plate.
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Add mushrooms, onion and wine to skillet and cook, stirring occasionally, until liquid has evaporated and mushrooms are browned, 8 to 10 minutes. Add flour and garlic; cook, stirring constantly, for 1 minute. Add broth and bring mixture to a boil, scraping up any browned bits. Add chicken, along with juices to skillet. Reduce heat to medium-low, cover and simmer until chicken registers 160 F, 5 to 10 minutes.
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Transfer chicken to clean platter and tent with foil. Whisk sour cream and egg yolk together in a bowl. Whisking constantly, slowly stir in ½ cup heat sauce into sour cream mixture to temper. Stirring constantly, slowly pour the sour cream mixture into simmering sauce. Stir in lemon juice, tarragon and nutmeg, return to simmer. Season with salt and pepper to taste, pour sauce over chicken and serve.