Defrost the rice cakes slowly in the refrigerator; then soak in water for about 4 hours.
Remove and discard the core and outer leaves of the cabbage and trim the leaves of the tough membranes. Blanch the cabbage in boiling, salted water for 1 minute until wilted, then shock in ice water. Roughly chop the cabbage and transfer to a food processor. Pulse until well chopped. Squeeze out excess water and transfer to a large bowl. Blanch the soy bean sprouts in the same boiling, salted water for 1 minute and shock in ice water. Transfer to a food processor and pulse until well chopped. Squeeze out the excess water and add to the cabbage. Stir in the chopped tofu. Transfer the mixture to a cheesecloth lined sieve, place a weighted plate on top of the mixture, and let stand over a bowl in the refrigerator for 2 hours to remove excess water.
Transfer the vegetable mixture to a large bowl and add the scallions, garlic, ground meat, salt, and sesame oil. Mix well.
To assemble the wontons, place a teaspoon of the filling in the center of a gyoza wrapper. Brush the edges with water and pinch the edges to seal. Repeat with the remaining filling and wrappers.
To make the broth, combine the flank steak and the water in a pot and let simmer for 1 hour. Remove the meat. When cool enough to handle, slice the beef thinly against the grain. Add the salt, soy sauce, and wonton soup base, to the broth. Drain the rice cakes, add them to the broth, and let simmer for 5 minutes. Add the wontons and sliced beef and let simmer for about 10 minutes more.
Garnish with the scallions, egg and nori.