1. Halve cabbage lengthwise. Cut halves crosswise into 1-inch-wide pieces. In a large bowl, toss cabbage with salt and 2 tablespoons sugar. Transfer to refrigerator and let stand overnight.

2. In a large bowl, combine garlic, ginger, kochukaru, fish sauce, usukuchi, shrimp, and remaining 1/2 cup sugar. If mixture is very thick, add water, 1/3 cup at a time, until consistency is just thicker than a creamy salad dressing. Stir in scallions and carrots.

3. Drain cabbage and add to bowl with garlic mixture. Cover and transfer to refrigerator for at least 24 hours. It is best when kept refrigerated for 2 weeks but will keep refrigerated for up to 1 month.