Preheat oven to 400°F. Stir pesto and garlic in medium bowl to blend. Transfer 1 tablespoon pesto mixture to small bowl and reserve for sauce.
Spread 1/4 cup pesto mixture over outside and 1 tablespoon inside main cavity of chicken. Place potatoes in large roasting pan and toss with remaining pesto mixture. Push potatoes to sides of pan; place chicken in center. Sprinkle chicken and potatoes with salt and pepper. Roast until thermometer inserted into thickest part of chicken thigh registers 180°F and potatoes are brown and tender, about 1 hour 20 minutes.
Transfer chicken and potatoes to platter. Add broth and reserved 1 tablespoon pesto to roasting pan. Place pan over heat; bring broth to boil, scraping up browned bits. Season sauce with salt and pepper. Serve chicken and potatoes with sauce.
