Heat a large Dutch oven or heavy pan to a touch less than medium heat. Pat the chicken very dry with paper towels and season very well with salt and pepper on both sides. Add the olive oil and begin searing the chicken skin side down. After 2 minutes, add 2 tablespoons of butter to the pan. Sear chicken for 5 minutes then flip and cook for another 3 minutes. Do not overcrowd the pot. Work in batches and place seared chicken onto a plate.
Add the remaining butter and the onions, celery, and carrots to the pan. Cook until soft (about 8-10 minutes) then add the mushrooms. Continue cooking until they release their liquid (about 5 minutes). Season the veggies with a touch of salt. Add the garlic and cook until fragrant (about 1 minute) then add the flour and cook for 1 more minute stirring frequently to thoroughly incorporate.
Add the wine and chicken stock to the pot. Turn heat to high and with a wooden spoon dislodge all of the flavor bits from the bottom of the pot. Cook for 1 minute then turn the heat down to medium-high.
Next, add the chicken and juices to the pot skin side up along with the parsley sprigs, thyme sprigs, and bay leaf. Cook for 5 minutes or until bubbling aggressively.
Turn heat down to medium-low and cover partially with a lid. Cook for approximately 30-35 more minutes or until the chicken thighs reach at least a 175f internal temperature.
Once the chicken is fully cooked, move it to a plate and finish the sauce. Turn heat to high and reduce the liquid in the pot by about half. This will take about 5 minutes. At this time remove the herb sprigs and bay leaf.
Whisk the egg yolks and heavy cream together in a bowl. Add 1 tablespoon of the hot cooking liquid while whisking. Continue to add the liquid 1-2 tablespoons at a time until you have added at least 1/2 a cup of the hot liquid. Pour the tempered mixture back into the pot and stir well.
Add the minced parsley into the sauce and stir to combine. Taste test the sauce and season with salt and pepper to taste. For a touch more acid, add lemon juice to your liking, but the wine is normally sufficient. Return the chicken to the pot and cook for 3-5 minutes or until the chicken is hot. Serve with crusty bread. Enjoy!