Cut cabbage into thin shreds, about 1/16 inch thick. This can also be done with a sharp knife or mandoline. Remove any large pieces and discard.
Work in 5-lb batches, combine shredded cabbage and 3 Tablespoons pickling salt in a large stone crock or glass or food-grade plastic container. Mix thoroughly. Let stand for 15 minutes or until juices start to flow and cabbage wilts slightly. Using a wooden spoon or your hands, press down firmly on the cabbage until the juice comes to the surface. Repeat four times, until all the cabbage is used up, leaving at least 4 inches of space between cabbage and rim of container. Sprinkle remaining pickling salt on top. If not enough juice has been produced to cover cabbage, add brine*.
Place a large clean inverted plate over the cabbage mixture and weigh down with two or three quart jars filled with water and capped. (Keep cabbage under brine by 1 to 2 inches throughout fermentation.) Cover with a clean heavy towel. Let stand in a cool place. Every day, remove and discard any scum that has formed. During fermentation, gas bubbles will form. When bubbling ceases, fermentation is complete. Fermentation will take from 3 to 6 weeks, depending on atmoshpheric conditions and variations in the cabbage itself.
*To make brine, combine 4-1/2 tsp pickling or preserving salt and 4 cups water. Bring to a boil over medium-high heat, stirring to dissolve salt. Let cool to room temperature, then ladle over sauerkraut to cover.
Prepare boiling water canner. Heat jars in simmering water until ready for use. Do not boil. Wash lids in warm soapy water and set bands aside.
Pack sauerkraut, with brine, into hot jars, leaving 1/2 inch headspace. Remove air bubbles and adjust headspace, if necessary, by adding brine. Wipe rim. Center lid on jar. Apply band until fit is fingertip tight. Place jar in boiling water canner. Repeat until all jars are filled.
Process quart jars in a boiling water canner for 25 minutes and pints 20 minutes, adjusting for altitude. Remove jars and cool. Check lids for seal after 24 hours. Lid should not flex up and down when center is pressed.