Fill a pot with about 2 liters of water. Add the ginger slices, whole scallions, and 1 tablespoon of salt and bring it to a boil.
Turn the heat down until the water is barely moving. Lower in the chicken breasts and hold them at that bare simmer for 5 minutes. Do not let it boil.
Turn off the heat, cover, and leave the chicken in the hot water for 20 minutes, until the thickest part reads 68C (155F).
Move the chicken to a bowl of ice water for 5 minutes. Keep every drop of the broth.
Cook the rinsed rice with 2 cups of the warm poaching broth, the smashed garlic, and the sesame oil, in a rice cooker or a covered pot.
Put the minced scallions, grated ginger, and half teaspoon of salt in a heatproof bowl. Heat the neutral oil until shimmering and pour it over. It should sizzle hard. Stir and set aside.
Slice the chicken against the grain. Serve over the rice with the sauce spooned on top and cucumber alongside. A small cup of the hot broth on the side is traditional and worth doing.