Cut cabbage lengthwise into quarters, keeping the leaves attached to the core. Rinse cabbage with cold water and liberally sprinkle 3 tablespoons salt between the leaves. Place cabbage in a large bowl and add just enough water to cover. Set aside until leaves are soft, 4 to 5 hours.
Rinse the salted cabbage 3 or 4 times with cold water and drain in a colander for about 20 minutes.
Combine 1 cup water, radish, green onions, garlic, ginger, sugar, 1 teaspoon salt, vinegar, and red pepper threads in a bowl. Sprinkle radish mixture between all the cabbage leaves except for the large outer leaf of each quarter. Reserve the liquid from the radish mixture.
Peel back the large outer leaf of each quarter, without removing it from the core, and fold the inner leaves in half. Wrap the outer leaf around the inner leaves and pack the cabbage quarters into a clean 1/2-gallon jar; pour in the reserved liquid from radish mixture. Seal jars and let sit at room temperature for 1 day. Slice the wrapped cabbage quarters into 1 1/2-inch pieces.