Start by thinly slicing the unpeeled carrots into matchsticks. Rinse and set aside.
Bring a large pot of water to a boil. Cook the pasta according to package directions.
While the pasta is cooking, melt 1 Tablespoon of butter over medium heat in a large skillet.
Add the thinly sliced carrots to the skillet with the melted butter and saute for 7-10 minutes or until the carrots become slightly caramelized and tender. Once done, remove them from the skillet and set aside.
Using the same skillet, add the olive oil and allow it to get hot. Add the minced garlic and saute for for 1-2 minutes.
Season the diced chicken with salt and pepper and add to the skillet. Cook until done, about 7-10 minutes or until chicken is no longer pink. Remove the chicken from the skillet and set aside.
When the pasta has finished cooking, drain and return the cooked pasta and 1 Tablespoon of butter to the skillet.
Add the pesto sauce, shredded parmesan cheese, sauteed carrots, and chicken to the large skillet with the pasta and toss to combine.
Divide the pasta among four plates. Garnish with fresh sliced basil. Serve immediately.