Finely grate enough zest from limes to measure 1 tablespoon. Cut limes crosswise in half, squeeze juice from limes. Place zest and juice in a blender; add cilantro, sugar, red chili pepper (if using), garlic, salt, and pepper. While processing, gradually add oil in a steady stream.
In a large bowl, combine cabbages, cucumber, edamame, avocados, carrots, and green onions. Drizzle with dressing; toss to coat.
Notes
English cucumbers have thin skins and few, if any, seeds. If you would like to use garden-variety cukes, peel the tough skin and, after halving lengthwise, scoop out the seeds with a spoon or melon baller.
