Season chicken on both sides with 1 tablespoon salt and 1/2 teaspoon pepper.
Preheat a Dutch oven or other large heavy pot over medium-high heat. Add 2 tablespoons butter and the oil to pot. When butter melts and foam subsides, add half the chicken, skin side down, in a single layer; do not crowd pot. (If butter begins to blacken, lower heat.) Fry chicken, turning once, until golden brown on both sides, about 10 minutes total, and transfer to a plate. Repeat with remaining chicken.
Reduce heat to medium, and add mirepoix (onion, carrot, and celery) to pot, scraping up any browned bits with a wooden spoon. Saute mirepoix, stirring occasionally, until soft and golden brown in places, 8 to 10 minutes.
Add mushrooms, and cook, stirring occasionally, until mushrooms darken, become glossy, and begin to release liquid, 4 to 5 minutes. Stir in flour, and cook until flour is absorbed by vegetables and is no longer visible, about 1 minute.
Add wine to pot, and bring to a boil, stirring until liquid just thickens, about 45 seconds. Add broth, and stir.
Place chicken, skin side up, in a single layer on vegetables; pour juices that have accumulated on plate into pot. Tie parsley sprigs, thyme sprigs, and bay leaf together with kitchen twine; add to pot.
Bring to a boil, then reduce to a simmer. Cover partially. Cook until internal temperature of thickest part of chicken registers 165 degrees, 25 to 30 minutes. Transfer chicken to a clean plate. Simmer liquid, uncovered, until reduced slightly, about 5 minutes. Discard herbs.
To make the liaison (sauce thickener), whisk together egg yolks and cream in a medium bowl. Whisking constantly, pour 1/2 cup cooking liquid, 1 tablespoon at a time, into liaison to temper it. Stir tempered liaison into pot.
Return chicken to pot. Add tarragon, lemon juice, and the remaining butter. Bring to a simmer, stir gently to combine, and serve.