Place chicken breasts, ginger, and scallion in a pot and cover with cold water. Bring to a gentle simmer over medium heat.
Reduce heat to low and poach chicken for 12 to 15 minutes until the internal temperature reaches 165°F (74°C).
Remove chicken from the liquid and submerge in an ice bath or let cool completely at room temperature.
Place cooled chicken on a cutting board and lightly pound with a rolling pin or meat mallet to loosen the muscle fibers.
Shred the chicken by hand into thin strips and set aside.
In a medium bowl, whisk together the sesame paste, soy sauce, black vinegar, chili oil, sugar, sesame oil, and ground Sichuan peppercorns until a smooth emulsion forms.
Arrange the julienned cucumber on a serving platter and layer the shredded chicken on top.
Drizzle the prepared dressing evenly over the chicken.
Garnish with toasted sesame seeds and chopped cilantro immediately before serving.