Bring the water to a boil in a large pot with the ginger slices, scallion whites, and salt. Add the chicken, making sure it is fully submerged. Return to a gentle boil for 1 minute, then turn off the heat, cover, and let stand for 22 to 25 minutes, until a thermometer in the thickest part reads 160°F (71°C).
Transfer the chicken to a bowl of ice water for 5 minutes, then drain and pat dry. Rub with the sesame oil and set aside. Strain the broth and keep it hot.
Heat the oil in a medium pot over medium heat. Cook the garlic and half of the grated ginger for 1 minute until fragrant. Add the rinsed rice and stir for 2 minutes to coat the grains.
Pour in 2 1/4 cups of the hot broth. Bring to a simmer, cover with a tight lid, reduce the heat to low, and cook for 15 minutes. Turn off the heat and rest, covered, for 10 minutes.
Blend the chilies, remaining grated ginger, lime juice, and 2 tablespoons of broth into a coarse sauce, with a pinch of salt to taste.
Slice the chicken against the grain. Serve over the rice with cucumber, the chili ginger sauce, and a drizzle of dark soy if you like.