Wash and dry carrots and green tops. Remove the feathery greens and set aside. Discard any tough stems.
Scrub carrots and dry with paper towel. If desired, peel the carrots. Cut into thin coins. Place the greens, carrots, nuts, basil leaves, salt, white pepper and cheese in the bowl of a food processor. Pulse to combine. With the processor on, drizzle in ½ cup olive oil. If you prefer a creamy pesto, add additional olive oil.
Store pesto in the refrigerator or freeze for later use.