Pulse carrots in a food processor until finely ground (almost puréed). Add chopped ginger, chopped shallots, rice vinegar, soy sauce, sesame oil, and salt and pulse until ginger and shallots are minced. With motor running, add vegetable oil in a slow stream.
Transfer mixture to a blender. Add 1/4 cup water and blend until smooth, 2 to 3 minutes. Thin dressing with additional water if desired.