Mise en place (aka prep work). Combine carrots, onions, and celery in a bowl. Mix, and set aside. In a large measuring cup, whisk together milk, chicken stock, ½ teaspoon salt, ¼ teaspoon pepper. Cover and set aside in the fridge.
Parboil potatoes. Peel and slice potatoes into ¼" rounds. Place in a pot with cold, salted water. Bring to a boil. Reduce to simmer, 5-10 minutes. When a fork can easily pierce through a slice, drain and set aside. The potatoes will not be fully cooked yet, that's okay.
Cook the chicken. Heat a large, high-sided skillet over medium heat. Add 1 tablespoon neutral oil. Pat chicken dry and season with salt and pepper on both sides.Once oil shimmers, place chicken in skillet and cook until browned, 3-4 minutes per side, then remove chicken from pan and set aside to rest on a cutting board. The chicken might not be fully cooked through yet, it will finish cooking later.
Cook the filling. Return skillet to medium heat, no need to clean it out. Melt 3 tablespoon butter, then add carrot, onion, and celery mixture. Cook until soft, about 4-5 minutes, stirring frequently. The moisture from the onions will loosen the brown bits (fond) on the bottom of the pan, so scrape those up as you stir — they add lots of flavor.
Thicken the filling. Add flour and stir constantly until lightly brown and no raw flour remains in the pan, about 2-3 minutes. If stuff starts sticking to the pan, add a small amount of oil (1-2 tsp) and continue stirring.Slowly add cold milk and stock mixture, stirring constantly until mixture reaches a low boil. Reduce heat and let simmer 3-5 minutes, stirring occasionally. Stir in additional stock if necessary to reach thick, gravy-like consistency.
Add chicken and potatoes. While filling is simmering, chop chicken into bite-sized pieces. Add back to skillet and stir to combine. Then stir in peas, tarragon, nutmeg, and additional salt and pepper (if needed) to taste. Finish by gently folding in the sliced potatoes so they retain their shape. Remove from heat until ready to use.