Core the cabbage and cut it into quarters lengthwise, then cut crosswise into 2-inch pieces.
In a large bowl, mix the cabbage with the salt and let it sit for 2 hours, turning it occasionally. After 2 hours have passed, rinse the cabbage thoroughly and drain well.
In a separate bowl, mix gochugaru, fish sauce, garlic, ginger, sugar, green onions, and carrots to create a paste.
Thoroughly mix the paste with the drained cabbage, ensuring even coating. It’s easiest to use your hands to do this, but you may want to wear disposable gloves.
Pack the mixture tightly into a clean jar, pressing down to remove air pockets. Leave about 1 inch of headspace at the top of the jar.
Seal the jar and let it ferment at room temperature for 1-5 days, depending on desired sourness. Then transfer to the refrigerator, where it will keep for up to 2 months.