1. Brown chicken: Season chicken on both sides with 1 tablespoon salt and ½ 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.
2. Sauté mirepoix: Reduce heat to medium and add mirepoix (onion, carrot and celery) to pot, scraping up any browned bits with a wooden spoon. Sauté mirepoix, stirring occasionally, until soft and golden brown in places, 8 to 10 minutes.
3. Cook mushrooms and flour: 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.
4. Pour in wine and broth: Add wine to pot, and bring to a boil, stirring until liquid just thickens, about 45 seconds. Add broth, and stir.
5. Simmer chicken: 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.
6. Make and add liaison: To make the liaison (sauce thickener), whisk together egg yolks and cream in a medium bowl. Whisking constantly, pour ½ cup cooking liquid, 1 tablespoon at a time, into liaison to temper it. Stir tempered liaison into pot.
7. Finish with tarragon, lemon and butter. Return chicken to pot. Add tarragon, lemon juice, and the remaining butter. Bring to a simmer, stir gently to combine, and serve.