Combine carrots, whole hazelnuts, garlic, and cheese in the bowl of a food processor; season with salt. Pulse until coarsely pureed. With motor running, slowly add oil, processing to a paste.
Bring a large pot of water to a boil; add salt. Cook pasta until al dente, according to package instructions. Drain; transfer pasta to a bowl. Add pesto and toss to coat evenly. Drizzle with oil, top with chopped hazelnuts, and serve immediately.