Soak the dangmyeon in warm water for at least 30 minutes, until pliable but not fully soft. Drain and set aside.
Bring a pot of water to a boil and blanch the chicken pieces for 2 to 3 minutes. Drain and rinse under cold water. This step removes surface impurities and keeps the final broth from turning cloudy or tasting gamey.
In a wide, heavy pot, combine the soy sauce, sugar, rice syrup, garlic, ginger and water or stock. Bring to a simmer.
Add the blanched chicken, cover, and simmer over medium-low heat for 15 minutes.
Add the potato and carrot. Continue simmering, covered, for another 15 minutes, until the chicken and vegetables are tender and the liquid has reduced by roughly half.
Add the onion wedges and sliced chilies (and gochugaru, if using). Simmer 5 more minutes, uncovered, to let the sauce keep reducing.
Add the drained noodles. Raise the heat slightly and cook, stirring gently, until the noodles are tender and the sauce has thickened to a glaze that coats everything with only a shallow, syrupy pool left at the bottom, about 5 to 7 minutes.
Take the pot off the heat, drizzle with sesame oil, and scatter scallions over the top before serving directly from the pot.