Remove excess fat and dry the chicken thighs and drumsticks with a paper towel, then season with salt and pepper.
Heat the butter and olive oil together in a deep skillet over medium heat, then cook the chicken thighs skin side down in the pan for 3-4 minutes without moving them. Flip them over and cook the other side for 3-4 more minutes or until they start to crisp up as well.
Remove the thighs from the pan and set them aside, then repeat the cooking process with the drumsticks. The chicken will not be cooked through at this stage, you just want to get some nice color on the skin.
Add the celery, carrots and onion to the skillet. Stir and cook for 5-6 minutes or until the veggies begin to soften.
Add the mushrooms, garlic, bay leaves and thyme. Stir and cook until the mushrooms start to caramelize, about 5 more minutes.
Add the flour and constantly stir as it starts to brown.
Pour in the chicken broth and white wine (if using), then bring the heat up to high and let the sauce come to a boil before reducing the heat and adding the chicken thighs and drumsticks back to the pan.
Cover the pan with a lid and let it simmer on low-medium heat for 45 minutes.
Open the lid, remove the chicken, pour in the heavy cream, and stir to combine, then let it simmer for 10 minutes and add salt and pepper to taste.
Add the chicken back to the pan, then serve garnished with freshly chopped parsley or fresh thyme.