Preheat the oven to 400°F.
Remove and discard the loose outer leaves from the sprouts, and cut the sprouts in half through the core.
Put the bacon in a 12" fry pan and cook over medium heat, turning occasionally until just about crisp, 5 minutes or so. With a slotted spoon, transfer the bacon to a paper towel-lined plate and reserve.
Drain off most of the fat from the pan and add the sprouts, cut side down in the same pan. Raise the heat to medium-high and sear until the sprouts begin to sizzle. Put the skillet in the oven and roast until the sprouts are deeply browned, 8 minutes or so, then shake the pan to redistribute them. Pull the pan from the oven when the sprouts are bright green and fairly tender (taste one to check), 10–15 minutes more.
Return the pan to the stovetop over medium heat and stir in the butter, bacon and salt and pepper to taste. Toss the sprouts to coat them.
Divide the kimchi among four shallow bowls. Use the back of a spoon to spread the kimchi out so it covers the bottom of the bowls. Divide the sprouts among the bowls, arranging them in a tidy pile on top of the kimchi. Garnish each with a pile of carrot julienne and serve.
Cut the cabbage lengthwise in half, then cut the halves crosswise into 1-inch-wide pieces. Toss the cabbage with the salt and 2 tablespoons of the sugar in a bowl. Let sit overnight in the refrigerator.
Combine the garlic, ginger, kochukaru, fish sauce, soy sauce, shrimp, and remaining 1/2 cup sugar in a large bowl. If it is very thick, add water 1/3 cup at a time until the brine is just thicker than a creamy salad dressing but no longer a sludge. Stir in the scallions and carrots.
Drain the cabbage and add it to the brine. Cover and refrigerate. Though the kimchi will be tasty in 24 hours, it will be better in a week and at its prime in 2 weeks. It will still be good for another couple week after that, though it will grow incrementally stronger and funkier.