Mix the ground chicken, beaten egg, panko, scallion whites, ginger and 1 teaspoon soy sauce in a bowl. Mix with your hand for about a minute until the mixture turns sticky and holds together. Wet your hands and shape into 8 oval patties, about 1.5 cm (1/2 inch) thick.
Stir the 2 tablespoons soy sauce, mirin, sake and sugar together in a small bowl until the sugar dissolves.
Heat the oil in a large nonstick skillet over medium heat. Add the patties and cook for 4 minutes without moving them, until the bottoms are deep golden. Flip and cook 3 minutes more.
Pour the tare over the patties, lower the heat slightly, and cover for 2 minutes so the chicken cooks through. Uncover and cook 2 to 3 minutes, turning the patties, until the sauce reduces to a glossy glaze and the centers reach 74C (165F).
Divide the rice between two bowls. Top each with four patties, spoon over the remaining glaze, and make a small well in the rice for an egg yolk.
Finish with the scallion greens, nori and sesame seeds. Break the yolk and stir it through the rice as you eat.