Put the cooked rice in a medium bowl. Heat the sesame oil in a large pot over medium-high heat until shimmering. Add the minced garlic and cook, stirring often, until the garlic is toasted and golden brown, 1 minute. Remove the pot from the heat and pour the garlic and sesame oil over the rice. Set the pot aside. Season the rice with 1/4 teaspoon salt and stir with a fork until combined, breaking up any lumps.
Stuff the cavity of each hen with 1/2 cup of the rice mixture. Overlap the skin flaps to enclose the rice inside the hens, fold the legs closed and tie with kitchen twine.
Place the hens breast-side down in the reserved pot and add the chicken broth, 2 cups water, shallots, ginger, scallion whites, smashed garlic cloves, sugar and 1 teaspoon salt. Bring to a boil over high heat, skimming the foam that rises to the top. Partially cover the pot and reduce the heat to maintain a steady simmer. Cook for 15 minutes, then use a spatula and a spoon to gently turn the hens breast-side up, being careful not to break the skin (they will float). Continue to simmer, partially-covered, until the hen breasts are firm and a thermometer inserted into the thickest part of the thigh registers 165 degrees F, 15 to 20 minutes more.
Meanwhile, make the sauces. Combine the soy sauce and vinegar in a small bowl and set aside. Combine the sesame oil, sesame seeds, 1/4 teaspoon salt and 1/4 teaspoon pepper in another small bowl and set aside.
Using a slotted spoon and tongs, remove the hens from the liquid, place them on a cutting board and remove the kitchen twine. Slice along the top of the breastbone to cut each hen into 2 halves, keeping the rice inside the cavity. Strain the broth through a sieve. Place each hen half in a wide, shallow bowl (rice side down) and divide the soup among the bowls. Garnish the soup with scallion greens and serve with the sesame sauce, soy-vinegar sauce, kimchi and seaweed.
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