In a large pot or Dutch oven, melt the unsalted butter over medium heat. Add the chopped yellow onion, celery, and carrots. Sauté until the vegetables are softened, about 5-7 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.
Toss in the bay leaves and diced Yukon gold potatoes. Stir everything together and season with a pinch of salt and pepper.
Sprinkle the all-purpose flour over the vegetable mixture, stirring constantly for about 1-2 minutes. This step helps cook off the raw flour taste and will thicken the soup later.
Slowly pour in the chicken broth while stirring to prevent lumps. Then, add the milk and bring the mixture to a gentle simmer.
Once the soup is simmering, stir in the shredded chicken and chopped fresh thyme. Allow the soup to simmer for about 15-20 minutes, or until the potatoes are tender.
In the last few minutes of cooking, add the frozen peas and corn. Taste and adjust seasoning with more salt and pepper if necessary. Remove the bay leaves before serving.
Ladle the soup into bowls and garnish with fresh chopped parsley. If desired, serve with pie crust crackers for a delightful crunch!