Peel and cut radish into ¾-inch cubes. Place radish cubes in a large bowl, toss with 1 tablespoon salt and up to 1 tablespoon sugar if using. Leave salted radish in bowl to draw out liquid for 10 minutes, up to 30 minutes.
Drain liquid from radish into a separate bowl and save.
Add garlic, ginger, green onions, gochgaru, and fish sauce to the radishes. Wearing gloves, mix the seasonings with the radish by hand, lightly massaging the seasonings into the radishes. If the seasonings are not distributing evenly, add little bit of the reserved salted radish liquid.
Transfer seasoned radishes to a large glass container with a tight-fitting lid like a glass jar. Gently tap the container against the counter and press down on the radishes with a ladle or your hands to remove and air pockets. Cover the container loosely with plastic wrap or with the lid, not sealed all the way shut.
To jumpstart the fermentation, you can leave the container of fresh kkakdugi on the counter overnight. Press down on the radishes into the liquid they release, then close the lid tightly and refrigerate.
You can eat kkakdugi right away; it will taste like spicy radish salad. After 1 day, it will be lightly fermented. After 3 days, it will taste like the kkakdugi you know and love.