**Sauté the Vegetables:** Melt butter in a large skillet over medium heat. Add the onion, garlic, carrots, and celery. Cook for 5-7 minutes until the vegetables are softened.
**Make the Sauce:** Sprinkle the flour over the vegetables and stir to coat evenly. Cook for 1 minute. Slowly pour in the chicken broth and milk, whisking constantly to prevent lumps.
**Season:** Stir in salt, pepper, thyme, and parsley. Bring the mixture to a simmer and cook until slightly thickened, about 2-3 minutes.
**Add the Noodles:** Stir in the egg noodles, ensuring they are submerged in the liquid. Cover the skillet with a lid and cook over medium heat for 8-10 minutes, stirring occasionally, until the noodles are tender.
**Add Chicken and Peas:** Stir in the cooked chicken and frozen peas. If desired, add heavy cream for extra richness. Cook for 2-3 more minutes, until heated through.
**Serve:** Remove from heat and serve hot, garnished with fresh parsley if desired.