To make the braised eggs, in a small pan that will snugly fit your eggs, add the tamari, sugar, water, mirin and chopped scallion, along with ½ cup of water. Bring to the boil, then reduce the heat to medium, add the hard-boiled eggs and simmer for 15 minutes, giving the pan a gentle roll around every few minutes to coat the eggs. Turn off the heat and scoop out the braising sauce – you will keep this for the noodles. Allow the eggs to cool.
Bring a large pot of salted water to the boil and add the sweet potato noodles. Cook according to the packet instructions for 2–3 minutes, until the noodles are just cooked. Drain and refresh under cold running water and, using kitchen scissors, cut the noodles so the strands are shorter and easier to eat. Set aside.
Place a wok or large frying pan over a high heat and add a big drizzle of oil. Add the onion, garlic and carrot to the wok, season with a pinch of sea salt and toss for 2 minutes. Add the mushrooms and cook for another 60 seconds. Next, toss in the noodles, spinach, sesame oil and about ½ cup of the reserved egg braising sauce and cook for 1–2 minutes, until the spinach is just wilted and everything is well coated in the sauce. Remove from the heat and add the shallots. Season with salt and pepper.
To serve, divide the japchae among plates and serve with the braised eggs on the side either halved or sliced up. Scatter over the sesame seeds.