For the Sauce:
3 whole jalapeño chilies, roughly chopped (see note above)
1 tablespoon aji amarillo pepper paste (see note above)
1 cup fresh cilantro leaves
2 medium cloves garlic
1/2 cup mayonnaise
1/4 cup sour cream
2 teaspoons fresh juice from 1 lime
1 teaspoon distilled white vinegar
Kosher salt and freshly ground black pepper
2 tablespoons extra virgin olive oil
For the Chicken:
1 whole chicken, 3 1/2 to 4 pounds
4 teaspoons kosher salt
2 tablespoons ground cumin
2 tablespoons paprika
1 teaspoon freshly ground black pepper
3 medium cloves garlic, minced (about 1 tablespoon)
2 tablespoons white vinegar
2 tablespoons vegetable or canola oil
For the Sauce: Combine jalapeños, aji amarillo (if using), cilantro, garlic, mayonnaise, sour cream, lime juice, and vinegar in the jar of a blender. Blend on high speed, scraping down as necessary, until smooth. With blender running, slowly drizzle in olive oil. Season to taste with salt and pepper. Sauce will be quite loose at this point but will thicken as it sits. Transfer to a sealed container and refrigerate until ready to use.
For the Chicken: Pat chicken dry with paper towels and place breast-side-down on a large cutting board. Using sharp kitchen shears, remove the backbone by cutting along either side of it. Turn chicken over and lay out flat. Press firmly on breast to flatten the chicken. For added stability, run a metal or wooden skewer horizontally, entering through one thigh, going through both breast halves, and exiting through the other thigh. Tuck wing tips behind back.
Combine salt, cumin, paprika, pepper, garlic, vinegar, and oil in a small bowl and massage with fingertips until homogenous. Spread mixture evenly over all surfaces of chicken
Light one chimney full of charcoal. When all the charcoal is lit and covered with gray ash, pour out and spread the coals evenly over half of coal grate. Alternatively, set half the burners of a gas grill to high heat. Set cooking grate in place, cover gill and allow to preheat for 5 minutes. Clean and oil the grilling grate.
Place chicken skin-side up on cooler side of grill with legs facing towards hotter side. Cover grill with vents on lid open and aligned over the chicken. Open bottom vents of grill. Cook until instant read thermometer inserted into deepest part of breast registers 110°F. Carefully flip chicken and place skin-side-down on hotter side of grill with breasts pointed towards cooler side. Press down firmly with a wide, stiff spatula to ensure good contact between bird and grill grates. Cover and cook until skin is crisp and instant read thermometer inserted into deepest part of breast registers 145 to 150°F, about 10 minutes longer. If chicken threatens to burn before temperature is achieved, carefully slide to cooler side of grill, cover, and continue to cook until done. Do not leave the lid off for longer than it takes to check temperature or chicken will burn.
Transfer chicken to a cutting board and allow to rest for 5 to 10 minutes. Carve and serve with sauce.