Combine minced garlic, olive oil, lime juice, aji amarillo or hot sauce, ground cumin, paprika, oregano, salt, and black pepper in a small mixing bowl to prepare the marinade.
For optimal cooking, spatchcock the chicken by removing the backbone or halve it using kitchen shears. Gently loosen the skin around the breast and thigh areas, applying half of the marinade beneath the skin and the remainder over the exterior. Place the chicken on a sheet pan, allowing it to marinate for at least 15 minutes while preheating the oven or grill.
Preheat the oven or grill to 400°F (190°C).
Oven Method: Roast the chicken in the oven for 30 minutes, basting it with its juices. Continue cooking for an additional 20 minutes or until the internal temperature reaches 165°F (74°C).
Grill Method: Lightly oil the grill grates. Position the chicken breast-side up over indirect heat and cook for approximately 50 minutes or until the internal temperature reaches 165°F (74°C).
While the chicken cooks, prepare the green sauce. Blend the cilantro, jalapeños, garlic, aji amarillo paste, olive oil, lime juice, vinegar, cheese, and salt in a food processor or blender until smooth. Gradually incorporate the mayonnaise by blending on low speed to preserve the sauce's creamy texture.
Once the chicken is cooked, allow it to rest for 10 minutes before carving. Serve with the green sauce, alongside cilantro lime rice, and garnish with lime wedges if desired.