In a bowl or liquid measuring cup, whisk together lime juice, 2 tablespoons olive oil, adobo sauce, garlic, cilantro, salt and black pepper.
Place the chicken in a large resealable plastic bag and pour in the marinade. Seal and turn the bag to make sure the chicken is coated. Place in the refrigerator for 30-60 minutes to marinate.
While the chicken is marinating, halve the potatoes and cut the broccoli into florets. Place the potatoes and broccoli in a bowl and add the remaining 1 tablespoon of oil along with a sprinkle of salt and pepper, tossing to coat.
Preheat oven to 425°F. If you like, line a rimmed baking sheet with parchment or foil.
Transfer the chicken to a sheet pan, discarding the marinade. Bake for 10 minutes, then remove the sheet pan from the oven and carefully spread the potatoes and broccoli around the chicken.
Bake for an additional 25-30 minutes, or until the chicken skin is crisp and the internal temperature reaches 165°F.