To prepare broth: Start the ramen soup early in the day or at least several hours before dinner.
Heat oven to 450 degrees. Put the carrots, onions, ginger and chicken in a cast-iron pan and sprinkle with salt and oil. Distribute the oil around to coat all the chicken and vegetables and roast for 30 to 45 minutes in the middle of the oven.
Scrape the roasted chicken and vegetables and all of the pan drippings into a large heavy pot with 4 quarts cold water and bring to a boil over high heat. Lower heat and simmer, covered, for 1 hour. Uncover, pull out 2 thighs, place in medium-sized bowl, and ladle a bit of broth over to allow meat to cool gently. Simmer stock, uncovered, for 1 hour more. After the chicken meat has cooled for a half-hour, shred, moisten with a ladle of broth, and reserve. When stock is done, strain into a clean saucepan and keep warm over low heat. (Discard remaining thighs and vegetables.)
To prepare noodle dough: Mix 2 tablespoons sesame oil into flour with your fingers until crumbly. Add eggs and egg yolks and stir with your hands until just incorporated. Knead on clean flat surface for 5 minutes until pliable but stiff. Let dough rest while you prepare ramen toppings.
To make half-boiled eggs: Carefully place eggs into boiling water and boil for 7 to 8 minutes. Plunge eggs into cold water and run more cold water over them until water temperature remains cool. Remove eggs from water when completely cool. Slice eggs in half vertically with razor-sharp knife. Scoop eggs from shells.
To finish noodles: Fill largest stockpot you own with hot water and bring to boil over high heat.