If the cabbage cores stick out too much or are dirty, trim them off with your knife. Tear away any outer leaves that are damaged or bruised.
To split a cabbage in half without shredding the densely packed leaves inside, first cut a short slit in the base of the cabbage, enough to get a grip on either half, and then gently pull the halves apart so the cabbage splits open.
Dunk the halves in a large basin of water to get them wet. Sprinkle salt between the leaves by lifting every leaf and getting salt in there. Use more salt closer to the stems, where the leaves are thicker.
Let the cabbages rest for 5-6 hours. If convenient, turn cabbages over every hour so they get well salted. From time to time you can ladle some of the salty water from the bottom of the basin over the top of the cabbages.
After 5-6 hours, wash the cabbage halves a few times under cold running water. As you wash, split the halves into quarters along the slits you cut into earlier. Cut off the cores, give them a final rinse, and put them in a strainer over a basin so they can drain well.