In a small bowl, combine garlic with olive oil, lime juice, aje amarilo or hot sauce, ground cumin, paprika, oregano, salt, and pepper.
Spatchcock the chicken, and gently loosen the skin at the breast and thighs areas. Using a small spoon, spread half of the paste under the skin, and the other half over the skin of the chicken. Place the chicken on a sheet pan, and let it sit with the marinade for 15 minutes as you preheat the oven or grill.
Preheat the oven or grill to 400°F/190°C.
Oven: Roast the chicken in the oven for 30 minutes, check on it and baste the chicken with its juices. Cook for 20 more minutes or until the internal temperature of the chicken reaches 165°F.
Grill: Oil the grill grates, then place the chicken breast side up over indirect heat. Cook for 50 minutes or until the internal temperature of the chicken reaches 165°F.
As the chicken is being cooked, make the cilantro sauce. In a food processor or blender, blend all of the ingredients except for the mayonnaise until smooth. Add the mayo, and blend again on LOW until just combined. (if you blend the mayo, you risk breaking the emulsion of it and it will separate).
Allow the chicken to rest for 10 minutes before carving. Serve with the sauce, cilantro lime rice, and lime wedges on the side.