Wash the cucumbers. Make pockets out of them by slicing them twice lengthwise at right angles, ½ inch from one end so the cucumber quarters are still connected to each other at the end. Put them in a big bowl, and sprinkle and rub them with salt inside and out. Let them sit for 10 minutes on the kitchen counter, and then turn them over so they're evenly salted. Then turn them over and let them sit another 10 minutes. Rinse the cucumbers in cold water a couple of times to remove excess salt. Drain and set aside.
Combine buchu (Asian chives), carrot, hot pepper flakes, fish sauce, sugar, and sesame seeds in a large bowl. Mix well with a wooden spoon.
Stuff the seasoning paste into the cucumber pockets. If you have sensitive skin, you can wear rubber gloves for this because the paste might irritate your hands.
Serve immediately as a side dish for rice, and keep the leftovers in the refrigerator. If you prefer it to be fermented a bit more, let the container sit at room temperature for a couple of days until it smells and tastes sour. Then put it into the fridge. Serve cold.
