In a large pot, melt the unsalted butter over medium heat.
Add the diced onion, carrots, and celery to the pot. Cook, stirring occasionally, until the vegetables are softened, about 7 minutes.
Add the minced garlic and cook for an additional 1 minute until fragrant.
Sprinkle the all-purpose flour over the vegetables and stir well to coat. Cook for 2 minutes, stirring constantly to avoid burning.
Gradually pour in the chicken broth while stirring to prevent lumps. Bring the mixture to a simmer.
Add the shredded cooked chicken, dried thyme, dried parsley, salt, black pepper, and bay leaf to the pot. Stir to combine.
Simmer the soup for 20 minutes, stirring occasionally, until the flavors meld and the soup thickens slightly.
Remove and discard the bay leaf.
Reduce the heat to low and stir in the heavy cream. Heat through for 3-5 minutes without boiling.
Taste and adjust seasoning with additional salt and pepper if needed.
Serve hot, garnished with fresh parsley if desired.