Heat the unsalted butter in a large skillet over medium heat.
Add the thinly sliced onions and cook, stirring occasionally, until caramelized and golden brown, about 20 minutes.
Add the minced garlic and cook for 1 more minute until fragrant.
Transfer the caramelized onions and garlic to the slow cooker.
Add the dried thyme, bay leaf, chicken broth, diced carrots, and diced celery to the slow cooker.
Place the boneless, skinless chicken breasts into the slow cooker, submerging them in the broth.
Season with salt and freshly ground black pepper to taste.
If using, pour the dry white wine into the slow cooker.
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 to combine.
Discard the bay leaf.
Taste and adjust seasoning with additional salt and pepper if needed.
Serve the soup hot, garnished with chopped fresh parsley.