Put the chicken breasts in a pot just big enough to hold them, add the ginger, scallions and salt, and cover with cold water by an inch. Bring it up to a bare simmer, then immediately drop the heat so the surface barely trembles. Poach gently for 10 to 12 minutes, until the thickest part reads 74C (165F).
Turn off the heat and let the chicken sit in the hot water for another 10 minutes. This is the step that keeps breast meat juicy, so do not skip it. Lift the chicken out and let it cool. Reserve a few tablespoons of the poaching liquid for the sauce.
Whisk the sesame paste with the soy sauce, black vinegar, sugar and grated garlic. It will seize up at first. Loosen it with 2 to 3 tablespoons of the warm poaching liquid, one spoon at a time, until it pours like thin cream. Stir in the chili oil, sesame oil and ground Sichuan peppercorn.
When the chicken is cool enough to handle, pull it apart along the grain into thick, rough shreds. Tearing by hand instead of slicing gives the sauce more surface to cling to.
Spread the julienned cucumber on a platter. Pile the shredded chicken on top and spoon the sauce over everything. Scatter with crushed peanuts, sesame seeds, extra scallions and cilantro. Serve at room temperature or chilled.