1. Pat chicken pieces dry and season generously with salt and pepper. Let rest briefly.
2. In a large skillet or heavy pot, melt butter (and olive oil if using) over medium-high heat. Brown the chicken thighs skin-side down for 4-5 minutes until golden, turn and cook the other side for about 1 minute, then remove. Brown drumsticks 2 minutes per side; remove. Avoid overcrowding; brown in batches if needed.
3. In the same pan with residual fat, sauté mushrooms, onions, bay leaf, and thyme for about 5 minutes until mushrooms turn lightly golden and onions soften. Add minced garlic; cook 30 seconds until fragrant.
4. Stir in flour, cooking for 1 minute to remove raw flour taste.
5. Deglaze with white wine, scraping the pan to lift browned bits. Simmer and reduce slightly, about 3-5 minutes.
6. Add chicken stock, season with additional salt and pepper. Return chicken pieces skin-side up to the pan, partially submerged in liquid.
7. Bring to a gentle simmer. Cover and cook for 10 minutes, then uncover and simmer an additional 20-50 minutes depending on desired tenderness, stirring occasionally until chicken is fully cooked and sauce thickens.
8. Remove chicken from pot. Stir in heavy cream and, if using, temper and whisk in egg yolks by slowly adding hot sauce liquid to yolks, then combine back into sauce. Warm gently without boiling to avoid curdling. Adjust seasoning to taste.
9. Return chicken to sauce, sprinkle with chopped parsley, and serve.