Hard-boil the eggs, then peel and cut in half. To make the chicken stock, place all the stock ingredients in a large pan, bring to the boil and simmer for 30 minutes, skimming off any impurities. Strain the stock into a clean pan big enough to take all the other ingredients, and set aside. The stock can be made a couple of days in advance and kept in the fridge.
Blanch the mangetout in a large pan filled with boiling water for 30 seconds, quickly scoop them out of the pan, reserving the water to cook the noodles, and plunge them into cold water to stop them from cooking. Drain and pat dry. Bring the water back to the boil, add the noodles and cook for about one minute if using fresh noodles, or longer if using dry noodles, following the packet instructions. Drain and refresh under cold running water.
When ready to serve, bring the stock to the boil and add the miso paste. Using a whisk, mix it vigorously so that the miso is completely dissolved in the stock. You may find it easier to mix the miso paste with a few tablespoons of the hot stock in a small bowl, and then add this to the pan. Now add the bok choy, red chillies and carrot as well as half the spring onion and coriander, and cook for one minute.
Add the cooked ramen noodles to one side of the pan to heat them through; it will only take a minute. Check for seasoning and add a little soy sauce if needed (miso saltiness varies from type used and from brand to brand). Just before serving, add the leftover roast chicken to a corner of the pan without mixing it into the noodles to warm it through.
Dish the ramen and vegetables equally among four bowls, topping each mound of ramen noodles with pieces of chicken, half a hard-boiled egg, the mangetout and the remaining spring onions and coriander. Drizzle a little toasted sesame oil over the dish, and serve with a generous sprinkle of chilli oil, white sesame seeds, shredded red chilli and a wedge of lemon.