1. Shave carrots into ribbons with a vegetable peeler to measure 2 cups. Combine carrots and onion in a large bowl.
2. Wash radishes and radish greens thoroughly; drain and pat dry. Cut radishes into thin slices to equal 1 3/4 cups; thinly slice radish greens to equal 1 cup. Add radishes, radish greens, basil, and next 4 ingredients (through pepper) to carrot mixture; toss gently to coat.
3. Arrange goat cheese on a serving platter; top with salad. Serve immediately.