Combine broth (or water), ginger, and the white parts of scallions in a wide skillet; bring to a gentle simmer.
Add chicken breasts, reduce heat to low, cover and poach 10–12 minutes until cooked through.
Remove chicken, let rest 5 minutes, then shred or thinly slice. Discard ginger slices or chop finely.
Toss cucumber with rice vinegar, sesame oil, soy sauce, and a pinch of salt; add honey if you want slight sweetness.
Arrange chicken over cucumbers, sprinkle sliced green scallions and red pepper flakes if using. Serve warm or slightly chilled.