Rinse the cabbages under cool water and remove the tough outer leaves. Cut the cabbages into quarters and cut out the core. Using a large, sharp knife, a food processor with a slicing blade or a mandoline, very thinly slice the cabbage. Place about one-third of the cabbage in a large clean bowl and sprinkle with 1 tablespoon salt; using clean hands, vigorously knead the salt into the cabbage until the cabbage starts to release a little liquid. Repeat with the remaining two-thirds of the cabbage and the remaining 2 tablespoons salt, vigorously kneading the salt into the cabbage after each addition.

Once all the cabbage is in the bowl, using both hands, massage the cabbage mixture vigorously, using your fingers to squeeze and bruise the cabbage, releasing as much of its liquid as possible, about 10 minutes.

Transfer the mixture and its liquid to a 5- to 6-quart glass, ceramic or stone container. Using your clean fists or a clean kitchen tool, pack the cabbage into the container with as much force as possible, removing all air pockets. Let stand, uncovered, for 2 hours. Pack the cabbage down once again. It should be completely covered in liquid. If not, add enough additional salt brine to cover.

Fill a sealable plastic bag (or bags) about two-thirds full with salt brine (instead of plain water in case they leak during fermentation). Place bag (or bags) directly on the surface of the cabbage mixture, using the bag(s) as water weights to keep the cabbage fully submerged at all times. Use enough water weights to cover the whole surface. Cover the container with a clean dish towel and place the lid on top. Place in a cool (60 degrees to 64 degrees F), dark place. The cooler the temperature, the slower the fermentation. If you want to speed up the process, place in a warmer spot, out of direct sunlight or heat.

Check the sauerkraut every few days. Remove any scum or bits of white/light gray mold from the surface with a clean spoon and wipe off the plastic bag as necessary. (White/gray mold is not harmful. If you see any black mold, discard the sauerkraut. Pink-colored "slime" on the surface is a yeast that, while not harmful, spoils the flavor and texture.) Pack the sauerkraut back down and replace the water weights. If the sauerkraut is not fully submerged, add additional salt brine. Replace the dish towel and lid.

After 2 to 3 weeks, use a clean fork to take out a sample to taste. If you like the flavor and have seen bubbles on the surface (a sign of fermentation), it's ready to be refrigerated. If you want more flavor, re-cover and continue fermenting until it develops a flavor that you like. When you like the taste, transfer the sauerkraut and liquid to smaller, airtight containers. Refrigerate for up to 6 months.