While water comes to a boil, thinly slice the English cucumbers into 1/8-inch rounds and set aside. Thinly slice the scallions and chop the cilantro, keeping them separate. Once the water is boiling, add the chicken breasts and 1.5 tsp kosher salt, returning to a boil. Reduce heat to low and simmer for 8-10 minutes until the chicken reaches an internal temperature of 165°F. Remove the chicken to a cutting board and let it cool for about 5 minutes—this brief rest keeps the meat from drying out when you shred it.
In a medium bowl, whisk together the seasoned rice vinegar, sugar, toasted sesame oil, ginger, red pepper flakes, and 1 tsp kosher salt. Stir until the sugar dissolves completely. I like to taste the dressing at this point and adjust the salt or vinegar to your preference—you want it balanced between tangy and slightly sweet with a gentle heat from the pepper flakes.
Once the chicken has cooled enough to handle, shred it by hand or cut it into bite-sized pieces, depending on your preference. Add the shredded chicken to the bowl with the dressing, then add the sliced cucumbers. Gently toss everything together until well coated. Let the salad sit for 2-3 minutes to allow the flavors to meld—the warmth of the chicken will slightly soften the cucumbers while they absorb the dressing.
Top the salad with toasted sesame seeds, sliced scallions, and fresh cilantro. Taste and adjust seasoning with additional salt and pepper if needed. I find that the sesame seeds add a nice toasted flavor and textural contrast, so don't skip them. Serve immediately on a bed of greens, alongside bread, or as a standalone salad.