Using a vegetable peeler, shave carrots lengthwise into very thin slices; place in a large bowl. Add oranges, red onion, beet, radishes, pistachios, tomatoes, capers, salt and pepper. Drizzle with vinaigrette; lightly toss to coat. Arrange lettuce on a platter; top with vegetable mixture. Top with cheese.
Can't find blood oranges? Use regular oranges or grapefruit.