To make the carrots, combine the rice vinegar, water, salt, and honey in a small pot and bring to a boil. Place the carrots in a heat-safe jar then cover with the hot liquid. Let sit for at least 10 minutes.
Remove the core from the cabbage cut into ¼” thick slices. Add sliced scallions and drizzle with oil, brushing/tossing everything together. Heat your broiler and roast the cabbage until charred. Don’t look away- it will go fast.
Measure out 3 tablespoons of the pickling liquid and combine with the soy sauce and sesame seeds.
Cook the noodles according to the package and finally, make the egg. Heat a small skillet over medium heat. Add the ghee and cook the eggs to your liking, spooning the ghee over the eggs to help the whites set.
Assemble the bowls by combining the noodles with the cabbage, some pickled carrots, the soy sauce mixture, and a solid sprinkle of sesame seeds. Top with an egg and serve with chili flakes and scallion greens if desired.