Lay a cucumber on a sturdy cutting board. Press the flat side of a heavy cleaver or rolling pin down firmly along its length until it splits into craggy pieces. Repeat with the remaining cucumbers, then tear or chop into bite-size chunks.
Toss the cucumber chunks with the kosher salt in a large bowl and let sit for 10 minutes. Drain off the pooled water and give the cucumbers a gentle squeeze.
Whisk the minced garlic, soy sauce, black vinegar, sesame oil, and sugar in a small bowl until the sugar dissolves.
Add the drained cucumbers to the dressing along with the sesame seeds and sliced scallions. Toss well so every craggy edge gets coated. Rest 5 minutes, then toss once more.
Taste a piece and adjust with more salt, soy sauce, or vinegar as needed. Serve cold or at cool room temperature.