To shred your cabbage, you can either shred it in a food processor using the shredding tool, or you can use a knife to cut the cabbage into thin ribbons. I prefer the food processor most days unless I'm feeling especially rushed when I'll buy a bag of pre-shredded cabbage at the grocery).
Preheat a medium saute pan over medium heat. Add your cabbage, followed by the vinegar, water, sugar, and caraway seeds.
Mix everything together and continue to cook over medium heat for about 5 minutes. At the 5 minute mark, turn the burner to low and continue to cook until the cabbage is fork tender. Try stirring every few minutes.
If you prefer your sauerkraut a bit more firm, you may want to stop the cooking process at the 5-minute mark. Personally, I prefer my sauerkraut to be a bit softer. Which means I need to cook the sauerkraut for around 10 to 15 minutes, total.
You can serve the sauerkraut warm with a roast or some broiled chicken thighs. Or allow it to cool and use it as a condiment for sandwiches for the week.
Store any leftovers in an airtight container for up to 5 days in your refrigerator (though I doubt it will last that long!).