Pat chicken dry with paper towels and season with 1 teaspoon salt and ½ teaspoon pepper. Heat butter and oil in 12-inch skillet over medium-high heat. When foaming subsides, place chicken in skillet and cook until browned, about 4 minutes. Flip chicken and continue to cook until browned on second side, about 4 minutes longer. Transfer chicken to large plate.
Add mushrooms, onion, and wine to now-empty skillet and cook, stirring occasionally, until liquid has evaporated and mushrooms are browned, 8 to 10 minutes. Add flour and garlic; cook, stirring constantly, 1 minute. Add broth and bring mixture to boil, scraping bottom of pan with wooden spoon to loosen browned bits. Add chicken and any accumulated juices to skillet. Reduce heat to medium-low, cover, and simmer until instant-read thermometer registers 160 degrees when inserted in breasts and 175 degrees when inserted in thighs, 5 to 10 minutes.
Transfer chicken to clean platter and tent loosely with foil. Whisk sour cream and egg yolk together in medium bowl. Whisking constantly, slowly stir ½ cup sauce into sour cream mixture. Stirring constantly, slowly pour sour cream mixture into simmering sauce. Stir in nutmeg, lemon juice, and tarragon; return to simmer. Season to taste with salt and pepper. Pour sauce over chicken and serve.