Wash cabbage and greens if needed, and cut into small, thin slices. Core and dice the apple. Combine cabbage, greens, apple, and carrot in a large bowl. Toss well.
To make the dressing, combine all ingredients in a small jar or liquid measuring cup. Whisk or shake until the mixture is smooth; dip a piece of cabbage or slaw into the dressing and sample; adjust salt, pepper, or other seasonings to taste.
Pour dressing over the slaw and toss to coat. Add pecans, toss gently once more, and serve.
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Notes
Using pre-shredded cabbage and greens saves a lot of time in this recipe. Another option I love is to use Trader Joe’s “Cruciferous Crunch,” which is a mix of shredded kale, Brussels sprouts, red and green cabbage. (If using this mix, I still add red cabbage to get the extra crunch and purple color.)
Tart, crisp apple varieties, such as Honeycrisp, work especially well in this slaw.
Walnut oil brings a wonderfully warm, rich flavor to this vinaigrette, and is well worth keeping on hand for this and other dressings. If you don't have it, however, you can absolutely substitute extra-virgin olive, avocado, or sunflower oil in this vinaigrette.