Heat the oil in a large deep skillet over medium-high heat. Season the chicken with 1 tsp salt and 1/2 tsp pepper. Brown the chicken in a single layer, turning once, about 5 minutes total. Transfer to a plate.

Reduce heat to medium. Add 1 tbsp butter, then the onion, carrot, and celery. Cook, stirring and scraping up browned bits, until softened, about 5 minutes. Stir in the garlic and cook 30 seconds.

Sprinkle the flour over the vegetables and stir for 1 minute. Gradually pour in the broth and milk while stirring, then add the thyme. Bring to a gentle simmer.

Return the chicken and any juices to the skillet. Add the egg noodles, stir, cover, and simmer, stirring once halfway, until the noodles are tender, about 8 minutes. Add a splash of broth if the sauce gets too thick before the noodles are done.

Remove from heat. Stir in the remaining 2 tbsp butter and a little extra milk if you like. Top with parsley and serve immediately.