How to Make It
Combine 6 cups water, beef stock, brisket, onion, and crushed garlic cloves in a Dutch oven; bring to a boil. Reduce heat to maintain a simmer; partially cover, and cook until brisket can be shredded (it will not be fall-apart tender, we want it with some chew), about 2 hours.
Place rice cake ovalettes in a large bowl, and add cold water to cover. Separate any that are stuck together. Let rice cakes soak 20 to 30 minutes. Drain.
Meanwhile, heat a medium-size nonstick skillet over medium. Combine egg yolks, 1 teaspoon water, and dash of kosher salt. Add 1/2 teaspoon sesame oil to pan, and swirl to coat. Add egg yolk mixture to pan, and swirl to form a thin layer; cook until just done on bottom without browning, about 1 minute. Flip egg, and cook just until done, 30 seconds to 1 minute. Turn egg out onto a cutting board. Roll up egg, and cut into thin slices.
Remove brisket from stock mixture; shred brisket with 2 forks, and place in a medium bowl. Add grated garlic, remaining 1/2 teaspoon salt, and remaining 1 teaspoon sesame oil; toss well to combine (I find this easiest to do with my hands).
Strain stock mixture through a fine wire-mesh strainer into a large bowl or 8-cup measuring cup; discard solids. Add enough water to stock mixture to equal 8 cups, and return to Dutch oven. Bring mixture to a boil. Stir in soy sauce. Add rice cakes, and cook until chewy-tender, 2 to 3 minutes. Return shredded brisket to soup, and remove from heat.
Ladle soup evenly into 6 bowls; top each with about 2 tablespoons sliced egg, 1 crumbled nori sheet, and 1 tablespoon scallion.