If you haven't already, prepare the Dressing recipe! For both the green and red cabbages, peel off any wilted, discolored, otherwise undesirable outer leaves. Quarter, core, and slice thinly, but not super duper thin! You don't want your slaw to become mush later after massaging it with the dressing.
Mix the green cabbage and carrots in a large bowl and place the red cabbage in another bowl. Split the dressing between the two bowls, and massage until the cabbage has softened — about 3 to 5 minutes. Note: before you dump ALL the dressing into the bowls, try reserving some. Massage with most of the dressing, and if it needs more/you like your slaw well-dressed, then put the rest in. Remember, you can always add more, as draining a soupy slaw is just kinda sad.
Cover both bowls, and refrigerate. Ideally, overnight, but give it at least an hour for the cabbage to soften/soak up the dressing.
Remove the bowls at least 10 minutes before serving. Mix the green and red together, and throw in the sesame seeds if you got them. If you're traveling with your cookout contribution and can/want to only take one bowl, put the green/carrot mix on top of the red, and mix once at your destination. [Transport the seeds in a small plastic bag/glass jar, which you can throw right on top of the cabbage in your large bowl.]
FOR FULL RECIPE