Heat olive oil in a large skillet over medium heat.
Add diced onion, sliced carrots, and sliced celery to the skillet and sauté for 5 minutes until vegetables start to soften.
Add minced garlic and sliced mushrooms to the skillet and cook for an additional 3 minutes until mushrooms release their moisture.
Sprinkle the flour over the vegetables in the skillet and stir well to coat. Cook for 1 minute to remove the raw flour taste.
Transfer the sautéed vegetables and flour mixture to the slow cooker.
Place the chicken breasts on top of the vegetables in the slow cooker.
Pour the chicken broth over the chicken and vegetables.
Add dried thyme, dried parsley, salt, and black pepper to the slow cooker. Stir gently to combine.
Cover and cook on low for 6 hours, or until the chicken is cooked through and tender.
Remove the chicken breasts from the slow cooker and shred them using two forks.
Return the shredded chicken to the slow cooker and stir in the heavy cream.
Cover and cook on low for an additional 15 minutes to heat through.
Taste and adjust seasoning with additional salt and pepper if needed.
Serve hot.