Preheat your oven to 350F. Heat your butter and oil in your oven proof Dutch oven and begin by lightly browning the chicken on both sides. Brown only a few pieces at a time, or they will steam rather than brown. Remove and set aside.
Add your mirepoix (onion,carrot and celery) into the pan and gently sauté as you scrape away any browned bits. These are packed with flavor.
Add your mushrooms and sauté until they give up their water. Add your flour and stir until the mixture starts to thicken. You are basically making a roux, to thicken the fricassee.
Add in your wine and cook until it reduces and begins to thicken. Add your broth and simmer gently. Then, add your thyme and bay leaf.
Put the chicken into the Dutch oven and gently stir to coat the chicken and into the oven uncovered for 40 minutes.
Just before the chicken is done, whisk together your egg yolks and heavy cream.
Remove the chicken from the oven and carefully add a few tablespoons of the sauce to the egg yolk/cream mixture, one at a time to temper it and then fold the mixture into the chicken to finish the sauce. Give it about one minute to thicken. Finish with lemon juice.