Salt the cabbage. Place the cabbage in a large bowl and sprinkle with the salt. Using your hands, massage the salt into the cabbage until it starts to soften a bit. Add enough water to cover the cabbage. Put a plate on top of the cabbage and weigh it down with something heavy, like a jar or can of beans. Let stand for 1 to 2 hours.
Rinse and drain the cabbage. Rinse the cabbage under cold water 3 times. Set aside to drain in a colander for 15 to 20 minutes. Meanwhile, make the spice paste.
Make the spice paste. Rinse and dry the bowl you used for salting. Add the garlic, ginger, sugar, and fish sauce, shrimp paste, or water and stir into a smooth paste. Stir in the gochugaru, using 1 tablespoon for mild and up to 5 tablespoons for spicy (I like about 3 1/2 tablespoons); set aside until the cabbage is ready.
Combine the vegetables and spice paste. Gently squeeze any remaining water from the cabbage and add it to the spice paste. Add the radish and scallions.
Mix thoroughly. Using your hands, gently work the paste into the vegetables until they are thoroughly coated. The gloves are optional here but highly recommended to protect your hands from stings, stains, and smells!
Pack the kimchi into the jar. Pack the kimchi into a 1-quart jar. Press down on the kimchi until the brine (the liquid that comes out) rises to cover the vegetables, leaving at least 1 inch of space at the top. Seal the jar.