Melt 2 tbsp. butter in a saute pan over medium heat. Add mushrooms, season with salt and pepper, and cook until browned and liquid nearly evaporates, 7-8 minutes. Transfer mushrooms to a plate.
Sweat onion, carrot, celery, and garlic in remaining 2 Tbsp. butter in same pan, covered, over low heat, until vegetables soften, 3-5 minutes.
Add broth, increase heat to medium-high, and bring to a boil. Stir in noodles, reduce heat to medium, and simmer until noodles are tender, about 10 minutes.
Combine half and half and flour in a jar, shake until smooth, and stir into broth. Add chicken, peas, and mushrooms; cook until sauce thickens and is bubbly, about 5 minutes. Stir in Thyme and Chives and season with salt and pepper and serve.