For the dressing: Pulse the carrots in a food processor until finely chopped. Add the ginger and onion and pulse until very finely chopped. Add the vinegar, miso, soy sauce, and 1/4 teaspoon salt; pulse until almost smooth. With the motor running, drizzle in the oil. Add a little water to thin the dressing slightly (about 3 tablespoons). Season with salt. Refrigerate until ready to use; the dressing can be made up to 2 days ahead.
For the salad: Toss the lettuce, cucumbers and cherry tomatoes with just enough dressing to coat. Serve immediately.