In a large Dutch oven or soup pot, melt butter over medium heat. Add onion, carrots, and celery. Cook, stirring occasionally, until softened, about 5-7 minutes. Add mushrooms and cook until they release their moisture and start to brown, about 5 more minutes.
Stir in garlic and cook for 1 minute until fragrant. Sprinkle flour over the vegetables and stir continuously for 2 minutes to cook out the raw flour taste.
Gradually whisk in the chicken broth, ensuring no lumps. Add wild rice blend, dried thyme, bay leaf, and nestle the whole chicken breasts into the liquid. Bring to a boil, then reduce heat to low. Cover and simmer for 30-35 minutes, or until rice is tender and chicken is cooked through (internal temp 165°F).
Remove cooked chicken breasts to a cutting board. Discard the bay leaf. Using two forks, shred the chicken into bite-sized pieces. Return shredded chicken to the pot.
Reduce heat to low. Slowly stir in the heavy cream and warm through, about 2 minutes. Do not boil. Season generously with salt and black pepper to taste.
Ladle soup into bowls, garnish with fresh parsley, and serve immediately.