Set aside 2 tablespoons of pesto in a separate bowl, for the sauce. Separate the skin gently from the breasts and thighs of the chicken by sliding a spoon under the skin. Stuff about ⅓ cup of the pesto under the skin with the spoon and massage it so it reaches as much of the chicken meat as possible. Sprinkle the chicken with salt and pepper. Slather with additional pesto.
In a roasting pan or large cast iron or stainless steel skillet, toss the potatoes with about 3 tablespoons of pesto until well coated. Move the potatoes to the sides of the pan and place the chicken in the middle.
Roast the chicken and potatoes in the preheated oven for about 60 to 90 minutes, depending on the size of your chicken. A meat thermometer inserted into the thickest part of the thigh should register 180F, and the juices should run clear when you separate the leg from the chicken.
Remove the chicken and potatoes from the oven and transfer to a serving platter. Allow the chicken to rest while you make the sauce.
Remove any black bits from the bottom of the pan and set over medium heat. Slowly pour the broth into the pan and whisk in the reserved 2 tablespoons of pesto. Bring to a simmer, whisking occasionally until the sauce is slightly thickened, about 5-10 minutes. Season with salt and pepper to taste and serve with the chicken and potatoes.