In a blender or a food processor, combine carrots, ginger, shallots, vinegar, miso, soy, sesame oil, and salt. Pulse to chop.
With processor running, slowly pour in oil. Puree until completely smooth, about 4 minutes. If dressing is too thick, slowly pour in water to thin to desired consistency.
Salad:
Divide chopped lettuce and cucumbers between serving bowls or plates.
Thinly slice remaining carrot into julienne strips. Place strips in a small bowl of ice water for 2 minutes so they curl a bit. Divide between servings. Drizzle with dressing and serve.