Heat the peanuts in the oil over medium heat, shaking them occasionally, until lightly browned after a few minutes. Blot with paper towels and set aside.
Using the vegetable peeler remove long strips of the carrot until you’ve reached the core.
Combine the cabbage; lettuce and carrot with everything except the nuts. Whisk the dressing ingredients together and toss with the greens. Add the peanuts just before serving.