Season the poaching water: Put the chicken thighs in a large pot or Dutch oven in a single layer, skin side up. Add the cold water, ginger slices, smashed scallions, Shaoxing wine, and kosher salt. The water should cover the chicken by about 1 inch; add a little more if it does not.
Poach, do not boil: Bring the pot to a bare simmer over medium-high heat — small bubbles around the edge, the surface just trembling. Immediately turn the heat down to low, cover, and cook for 12 minutes. The water should never reach a rolling boil.
Rest in the hot liquid: Turn the heat off and leave the covered pot alone for 10 minutes. Check the thickest thigh with a thermometer (165°F / 74°C) or by cutting into the thickest spot — the juices should run clear.
Make the sauce while the chicken rests: Put the sliced scallions, minced ginger, salt, sugar, and white pepper in a heatproof bowl. Heat the neutral oil in a small saucepan over medium-high heat until it shimmers and a scallion slice dropped in sizzles at once, about 350°F (175°C). Pour the hot oil straight over the scallion mixture, stir in the toasted sesame oil, and taste for salt.
Lift, rest, and slice the chicken: Move the chicken to a cutting board and let it rest 5 minutes. Slice across the grain into 1/2-inch pieces and arrange them on a warm plate, slightly overlapping.
Sauce and serve: Spoon about half the warm ginger-scallion sauce over the sliced chicken and serve the rest in a small bowl at the table. Serve with steamed jasmine rice and the cucumber slices if using.