To prepare the cucumbers, in a bowl, cut off the ends, quarter the cucumbers lengthwise, then slice into 2-inch pieces, sprinkle with the coarse sea salt, and toss to coat evenly, then let rest for 20 minutes and discard the released liquid.
To make the kimchi sauce, in a bowl, combine the salted shrimp, garlic, ginger, gochugaru, fish sauce, and plum extract, and mix until well combined.
To mix the kimchi, add the sauce to the cucumbers and toss well to coat evenly, then add the garlic chives and carrot and mix again until everything is combined.
To ferment, transfer the cucumber kimchi to an airtight container, let sit at room temperature for about 2 hours, then move to the refrigerator.
To serve, enjoy right away or after chilling, when the flavors have developed further.