Using the shredder blade on your food processor, finely shred the carrot, put it in a container, cover it, and refrigerate until needed.
You don't need to clean the food processor between shredding the carrots and shredding the cabbage. The two ingredients get put together later anyway.
Remove the outer leaves of the cabbage. Wash it. Cut it in half and remove the thick inner core. (Refrigerate the other half of the cabbage for another recipe.)
Using the shredder blade on your food processor, finely shred the cabbage.
Sprinkle the shredded cabbage with salt. Mix to combine.
Place the cabbage in a colander with a lightweight towel on top. Place a heavy plate or some unopened cans of vegetables on the top. This will help press the moisture out.
Put the colander in a bowl to catch the moisture, and set it aside for at least an hour (up to 4 hours).
When the time is up, rinse the cabbage thoroughly and pat it dry with a clean towel.
In the meantime, make the vinaigrette.
Mix the sugar, ginger paste, red pepper flakes and vinegar together until the sugar is dissolved.
Pour the vinaigrette over the cabbage mixture and stir to combine.
Cover the salad and refrigerate for at least 6 hours (longer if you can wait...it only gets better as it marinates).