In a large bowl combine the cabbage, apple, the 1 Tbsp. salt, and the caraway seeds. Using clean hands, massage the mixture for 1 to 2 minutes. Cover and let stand 2 hours, massaging it once or twice during this time.
Using clean hands, transfer the cabbage to a wide-mouth 1-qt. jar, packing it down using a muddler or wooden spoon. Pour any liquid remaining in the bowl into the jar over the cabbage.
For brine, combine the 1 cup water and the 1 tsp. salt. Pour enough of the brine into the jar so the cabbage mixture is covered by 1/2 to 1 inch. Use a cabbage leaf slightly bigger than the jar opening to place over the shredded cabbage, to keep it submerged under the brine. Cover jar using a recap airlock lid or other airlock lid. Reserve remaining brine to add, if needed, during fermentation.
Let stand at room temperature (no warmer than 75°F) for 3 to 4 weeks. In temperatures between 60°F and 65°F, fermentation will take 5 to 6 weeks. The sauerkraut is ready when it has a slightly crunchy texture and pleasantly tangy flavor. Place a plastic lid on the jar and transfer to the refrigerator. Chill 1 week before serving. Store in the refrigerator up to 2 months.
Use a mandoline to thinly slice cabbage.
(0 g saturated fat,
0 g polyunsaturated fat,
0 g monounsaturated fat),
0 mg cholesterol,
397 mg sodium,
4 g carbohydrates,
1 g fiber,
2 g sugar,
1 g protein.