To prepare the cucumbers, cut off both ends of the cucumbers. Slice them lengthwise in a cross pattern without cutting all the way through and transfer them to a mixing bowl.
In a pot over high heat, combine the water and Korean coarse sea salt. Stir until dissolved and bring to a boil. Turn off the heat and carefully pour the hot brine over the cucumbers. Let them brine for 30 minutes.
Rinse the cucumbers with water and drain well.
To make the stuffing, in a mixing bowl, combine the garlic chives, onion, carrot, garlic, gochugaru, fish sauce, salted shrimp, plum extract, sugar, sesame seeds, and water. Mix well.
To stuff the cucumbers, gently open each cucumber and fill the cuts with the stuffing. Coat the outside of the cucumbers with the stuffing as well. Transfer to an airtight container.
To ferment the cucumbers, you can enjoy them immediately, or leave them at room temperature for about 1 day to develop a tangier flavor.
Transfer to the refrigerator and continue fermenting to your taste. For the best balance of flavor and crunch, enjoy within 1 to 2 weeks.