1 1/2 pounds Kirby cucumbers, Korean cucumbers, or any other unwaxed pickling cucumbers
1 tablespoon + 1 teaspoon kosher salt, divided
2 teaspoons grated garlic
1 teaspoon grated ginger
1 to 2 tablespoons Korean red pepper flakes (gochugaru)
1 teaspoon sugar
4 ounces Korean radish or daikon, peeled and julienned
2 ounces sliced Korean chives or scallions, trimmed and cut into 1-inch pieces
Sesame seeds for serving (optional)
Wash the cucumbers and pat them dry. Trim and discard the ends. Cut the cucumbers in half for Kirbys or into about 4 pieces for longer Korean cucumbers (each piece should be about 2 to 3 inches long). Stand each piece upright on a cutting board, then slice it in quarters, leaving about 1/2 inch intact at the bottom end. Place the cucumbers cut-side up in a large dish and evenly sprinkle 1 tablespoon of salt over them. Let stand for 30 minutes.
Meanwhile, make the filling. Combine the remaining 1 teaspoon of salt, garlic, ginger, red pepper flakes, and sugar together in a bowl to make a thick paste. Using your hands, mix in the radish and chives or scallions and thoroughly coat them with the paste. (You may want to wear gloves to protect your hands.)
Drain the cucumbers and pat them dry. Stuff each cucumber with the filling, making sure to fill each crevice while being careful not to break the cucumber apart.
The cucumbers may be served immediately, chilled first, or fermented and then chilled (recommended). For fermented cucumbers, nestle the cucumbers together in a very clean jar or another container. Cover tightly and let stand at room temperature for 1 to 24 hours. Transfer to the refrigerator and eat within one week. Serve as is or sprinkle sesame seeds on top.