Cut the cabbage into bite-size pieces first — roughly 3 to 4 cm squares — before it touches any salt. Small pieces salt through in a fraction of the time whole leaves need.
Toss the cut cabbage with the salt in a large bowl, massaging it in lightly, and let it sit 25 to 30 minutes, tossing once halfway through.
Rinse the salted cabbage twice in cold water and drain well in a colander, pressing gently to remove excess water.
While it drains, stir together the gochugaru, fish sauce, sugar, garlic, and ginger in a small bowl to make the seasoning paste.
Combine the drained cabbage, scallions, and carrot in a large bowl. Add the paste and toss by hand — gloved, if you'd rather not smell like chili for the rest of the day — until every piece is coated.
Pack the seasoned kimchi into a clean container, pressing down so the liquid it releases rises to cover the solids.
Leave it at room temperature for 6 to 12 hours to start fermenting, then refrigerate. It's ready to eat within a few hours, and better after a day or two.