In a large bowl, whisk together mayonnaise, sour cream, vinegar, shallot, Worcestershire, 1/8 teaspoon pepper, and 1/4 cup cold water. Stir in 1/2 cup cheese, breaking up any large pieces with a fork; set aside.
With a vegetable peeler, peel carrots into long, thin ribbons; add to bowl with dressing along with radishes and lettuce. Toss to combine; season lightly with salt. Divide among four bowls; sprinkle with remaining 1/4 cup cheese. Serve immediately.