Melt 1 tbsp butter in a large pot over medium-high heat. Add the chicken, season with a pinch of salt and pepper, and cook until no longer pink outside, about 5 minutes. Remove with a slotted spoon and set aside.

Add another 1 tbsp butter to the pot. Scatter in the mushrooms in a single layer and cook undisturbed for 3 minutes, then stir and cook until deeply browned, another 4 to 5 minutes. Transfer to the plate with the chicken.

Reduce heat to medium, add the remaining 1 tbsp butter, onion, carrots, and celery. Cook for 5 minutes until softened, then stir in garlic for 30 seconds. Sprinkle in the flour and stir constantly for 1 minute. Slowly whisk in the broth and milk, then add thyme, salt, pepper, and the reserved chicken and mushrooms.

Bring to a gentle simmer, then reduce heat to low. Cook for 15 to 18 minutes, stirring occasionally, until the carrots are tender. Taste, adjust salt if needed, then ladle into bowls and top with parsley.