Grate the onion, garlic, and ginger on the smallest holes of a box grater, on a microplane, or process into a chunky puree in a food processor.
Combine onion/garlic/ginger with the remaining ingredients except the salt and additional water. Taste for seasoning and add salt if necessary. If you like the consistency thinner like ketchup rather than thicker like aioli, stir plain water into the Bibim Naeng-Myun sauce ½ teaspoon at a time until you get the consistency you like.
Cook Noodles
Right before serving, cook the naeng-myun noodles according to the package. Bring water in a large pot to a boil. Turn down the heat to medium and let cook for about 5 minutes. Drain into colander and rinse with cold running water several times while gently "massaging" the noodles.
Drizzle the rinsed and drained noodles with sesame oil and gently massage the noodles to distribute the oil.
Prep Naeng-Myun Bowls/Serve
Divide the cooked naeng-myun noodles among serving bowls right away because the noodles are somewhat "sticky" and start to clump together fairly soon (if they are left without liquid/broth).
Top the noodles in each bowl with 3-4 tablespoons of Bibim Naeng-Myun Sauce — the thick sauce will help to "anchor" the other ingredients on top. Add the julienned Asian pear, cucumber, and slices of half a hard boiled egg.
Add 2-3 chicken stock ice cubes to the bowls if you're using it. As the stock melts, it will mix with the thicker Bibim-Naeng-Myun red sauce and make it easier to mix the noodles, vegetables and sauce together in the bowl.
Garnish with toasted sesame seeds and chopped scallions. Serve with additional Bibim Naeng-Myun Sauce, vinegar, and kimchi on the side.