Whisk dressing ingredients in a medium bowl until smooth. Taste and adjust to personal preference. If desired, add in 1-1/2 tablespoons lemon juice (add slowly, to personal preference). Refrigerate while prepping the slaw.
Remove any wilted outer leaves. Cut cabbage into quarters, remove cores, and slice very thin. Cut long strands in half if needed. Wash and dry completely using a salad spinner. Repeat with red cabbage. Grate carrots on the large holes of a box grater. Cut apple into thin matchsticks.
Add cabbage, carrots, and apple to a large bowl. Pour dressing over and toss well until evenly coated. It may seem dry at first, but keep tossing. Cover and refrigerate at least 30 minutes, preferably 2 hours or up to overnight. Toss again before serving.
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Notes
Note 1: If using the apple, cut into thin matchsticks and place in a bowl. Cover with water and 1 teaspoon salt, gently stir, and let sit 10 minutes. Drain, rinse with cold water, and dry well. This will keep it from browning.Make ahead: This coleslaw is best after chilling. The cabbage softens slightly and the flavor deepens.Storage: Store in an airtight container in the fridge for 3 to 5 days. Do not freeze.