Heat the oven to 400°.
Line a baking sheet with parchment paper.
Peel and trim the ends off the carrots. Chop into 1" pieces and add to a large bowl.
Scrub the potatoes under running water and dry. Chop into 1" pieces and add to the bowl.
Toss the veggies with 2 tablespoons of oil, 2 teaspoons of salt and 1 teaspoon of black pepper until thoroughly coated.
Spread out on the baking sheet in a single layer, making sure there's enough in the middle to rest the chicken on.
Pat the chicken dry and trim off any excess fat around the cavity. Wiggle your finger between the skin and meat to loosen the skin. You don't want the skin to rip but don't worry if it does.
Smear the pesto under the chicken skin. Use your fingers and the skin to evenly spread out the pesto as best as possible. You should have about 1-2 tablespoons remaining pesto which you can spread out over the skin.
Cut the lemon in half and stick in the cavity.
Place the chicken on top of the vegetables with the legs facing the long side of the baking sheet. Place in the oven with the legs toward the back of the oven.
Roast until the skin is crispy and the vegetables are tender. An oven probe or instant thermometer should register 165° when inserted into the thickest part of the breast (without hitting bone). You can also cut into the chicken at the thigh joint - the juices should run clear. This should take about 60-75 minutes, depending on the size of the bird.
Remove from the oven, loosely cover with foil and let rest for 15 minutes. Remove the lemon from the cavity. Remove the chicken to a cutting board and carve. Toss the veggies on the baking sheet to get evenly coated with the juices. Serve the chicken and veggies, squeezing some lemon over everything.