Thinly shred the green and purple cabbage so pieces are roughly 1/8" wide. Julienne or grate the carrot. Thinly slice the green onion. Place all prepared vegetables in a large mixing bowl and toss to combine.

If you prefer a slightly softer slaw, sprinkle 1/2 teaspoon salt over the cabbage mixture, toss, and let sit 10 minutes; then gently squeeze out any excess liquid and return to the bowl.

Dressing

Make dressing

In a medium bowl or jar combine the mayonnaise, apple cider vinegar, lime juice, honey, Dijon mustard, olive oil, soy sauce, toasted sesame oil, grated ginger, minced garlic, salt, and black pepper. Whisk or shake vigorously until smooth and emulsified. Taste and adjust: add more honey for sweetness, vinegar or lime for brightness, or a pinch more salt if needed.

If you want a lighter dressing, substitute 1/4 cup mayonnaise + 1/4 cup plain yogurt and whisk the same way.

Assemble & finish

Combine

Pour about three-quarters of the dressing over the prepared cabbage mixture and toss thoroughly with tongs or salad servers until all vegetables are coated. Add additional dressing as needed to reach desired creaminess — reserve a small amount to adjust at the table if serving later.

Fold in the chopped cilantro (if using) and sliced jalapeño (if using). Sprinkle the toasted sesame seeds and chopped nuts over the top and gently toss once to distribute.

Let the slaw rest in the refrigerator at least 20 minutes before serving to allow flavors to meld. It keeps well for 1–2 days refrigerated; toss again before serving and adjust seasoning if needed.