In a Dutch oven or large stockpot set over medium heat, melt the butter. Incorporate the mushrooms, celery, onion, and carrots, and sauté the mixture, stirring occasionally, until softened, approximately 8 to 10 minutes. Blend in the garlic and thyme, continuing to cook until aromatic, around 1 minute. Add the chicken breasts, chicken stock, wild rice, bay leaf, along with 1 teaspoon of kosher salt and 1/2 teaspoon of ground black pepper. Elevate the mixture to a boil, then reduce the heat to low and cover. Allow it to simmer gently until the chicken is fully cooked and the rice has achieved a tender consistency, which should take between 35 and 45 minutes. Remove the chicken and set it aside until cool enough to handle. Discard the skin and bones, and dice the meat into bite-sized pieces. Return the chicken to the pot, incorporate the parsley, and adjust the seasoning with additional salt and pepper as needed. Serve hot.