Roughly chop one head of cabbage, removing the core, and put it into a food processor. Pulse until the cabbage is finely chopped, about 1/4 inch sized pieces or less. Pour into a large mixing bowl.
Peel carrot and cut into one inch pieces and put them in the food processor (no need to wipe out the processor bowl). Pulse until the carrot is finely chopped, about the same size as the cabbage, and add the carrot to the mixing bowl.
Peel the quarter onion, half an onion if it's a smaller size, and add it to the food processor. Pulse until the onion is well minced, scraping down the processor bowl as needed. Using a small food chopper is often easier for this than a full sized food processor. Add the minced onion to the mixing bowl.
Mix the cabbage, carrot and onion in the bowl until well blended and set aside while you make the dressing.