Preheat to 400F with the oven rack in the center of the oven.
Add the garlic, cumin, olive oil, paprika, pepper, oregano, 1/2 tsp salt, lemon zest, and 2 tbsp lemon juice to a small bowl.
Butterfly the chicken by cutting out the backbone with kitchen shears, and cracking the breastbone with your palms to flatten the chicken. Clean and pat dry.
Rub the chicken with your 2 leftover lemon quarters squeezing the juice on the chicken and rubbing the whole bird with the rinds. Take your spice mixture and rub it all over the chicken including under the skin. Be careful not to rip the skin.
Sprinkle the chicken with the remaining 1 tsp salt.
Transfer the chicken, butterflied, skin side up, to a sheet pan.
Roast 20 minutes. Baste. Roast 20 minutes. Baste. Roast 10-20 minutes until cooked. Juices should run clear and/or internal temperature in the thickest part of the thigh should read 165F.
While your chicken is in the oven, make your picquillo pepper pesto: Add all ingredients to a food processor. Yup, that's it!