Trim the ends off each cucumber. Standing each one on end, cut two deep slits in a cross pattern from one end, stopping about 2 cm / 3/4 in before the bottom so the four quarters stay attached.
Rub 3 tbsp salt over and into the cut cucumbers. Let them sit 20 to 25 minutes, turning once, until they've softened slightly and released visible liquid.
Rinse the cucumbers briefly under cold water and stand them upright in a colander to drain for 10 minutes.
Stir together the gochugaru, fish sauce, sugar, garlic, and ginger in a bowl to make a thick paste.
Fold in the scallions, garlic chives, and carrot.
Holding each cucumber over the bowl of filling, gently pack the paste into all four slits until it's visibly stuffed inside, not just smeared on the surface.
Arrange the stuffed cucumbers snugly in a container, any leftover filling spooned over the top.
Leave at room temperature for 3 to 6 hours, then refrigerate. It's ready to eat the same day and best within the first week.