Rub 1 side of nori with 1/2 teaspoon sesame oil; broil, oiled side up, 30 seconds. Turn nori; broil 30 seconds or until browned (or a dark green color). Crumble nori; set aside.
Heat vegetable oil in a large Dutch oven over medium-high heat. Add ginger and garlic, and stir-fry 30 seconds. Add chicken, and stir-fry 4 minutes. Add water, soy sauce, and broth, and bring to a boil. Cover, reduce heat, and simmer 15 minutes.
Soak rice cake slices in cold water 10 minutes; drain. Add to broth mixture. Increase heat to medium; cook, uncovered, 5 minutes or until rice cake slices are tender.
Reduce heat to low. Slowly drizzle egg into soup, stirring constantly. Cook 1 minute, stirring constantly. Stir in onions and 1 teaspoon sesame oil. Sprinkle with nori and sesame seeds.