Remove the carrot tops and soak in a bowl of cold water to clean. Rinse and clean carrots and set aside. Drain the carrot tops and pat dry with a paper towel.
Finely chop the carrot tops and add to a small bowl. Add the finely chopped garlic, walnuts, and olive oil and stir to combine. Salt and pepper to taste. Set aside.
Using a sharp paring knife or a vegetable peeler, peel the carrots into thin ribbons and transfer to a bowl of cold water. Discard tough carrot cores and the remaining tops. Let the ribbons soak for a minute or two (this helps them stay crisp and bright). Drain the ribbons and gently pat dry with a paper towel. Return to bowl and top with carrot top pesto, tossing to combine. Add a splash of red wine vinegar and toss again. Serve immediately.