Begin by organizing your workspace and prepping all ingredients. Dice your vegetables uniformly (about ¼-inch pieces). Season your chicken with salt and pepper on both sides.
Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add chicken breasts and cook until golden brown, about 4-5 minutes per side. They don't need to be cooked through completely. Remove to a plate and set aside.
In the same pot, add diced onions, carrots, and celery. Cook for 5-6 minutes until they begin to soften. Add minced garlic and mushrooms, cooking for another 3-4 minutes until mushrooms release their moisture and begin to brown.
Stir in the wild rice blend, then add chicken broth, bay leaves, thyme, and rosemary. Return the chicken to the pot. Bring to a boil, then reduce heat to maintain a gentle simmer. Cover and cook for about 35-40 minutes, or until rice is tender and chicken is fully cooked.
Remove the chicken and shred it using two forks. In a separate saucepan, melt butter over medium heat. Whisk in flour and cook for 1-2 minutes to create a roux. Gradually whisk in milk until smooth, then cook until thickened, about 5 minutes.
Remove bay leaves from the soup. Stir the creamy mixture into your soup pot, then add the shredded chicken and heavy cream. Simmer gently for 5-10 minutes to meld flavors. Adjust seasoning with salt and pepper to taste.
Serve hot, garnished with fresh chopped parsley.