Season chicken with salt and pepper. Melt 2 tbsp butter in a Dutch oven over medium-high heat and brown the chicken skin-side down for about 7 minutes, then flip and cook 5 minutes more. Transfer to a plate.

Reduce heat to medium, add remaining 1 tbsp butter, and sauté onion, carrots, and celery for 5 minutes until softened. Stir in garlic and cook 1 minute.

Sprinkle flour over vegetables and whisk constantly for 2 minutes. Slowly whisk in chicken broth and milk, bring to a simmer, then add thyme sprigs.

Stir in noodles, return chicken to the pot skin-side up, and lower heat to a low simmer. Cover partially and cook 18-20 minutes, stirring once halfway, until noodles are tender and chicken is cooked through.

Remove chicken and thyme stems. Shred chicken off the bone, return it to the pot, and stir. Ladle into bowls and top with parsley.