Prepare the Marinade: Combine olive oil, lime juice, lime zest, garlic cloves, cilantro, salt, paprika, black pepper, cumin, oregano, and sugar in a blender and blend until completely smooth.
Marinate the Chicken: Clean the chicken and pat dry with paper towels. Using your fingers, gently pull the skin from the flesh over the breasts and legs without tearing it. Spread most of the marinade underneath the skin directly onto the flesh, and rub the remaining marinade over the skin. Place in a Ziploc bag and refrigerate for at least 6 hours, or preferably overnight.
BGE Setup: Set the Big Green Egg up for rotisserie cooking using your rotisserie kit. Preheat to 350°F, pushing a small pile of lump charcoal to the back of the firebox to create an indirect zone.
Load the Rotisserie: Truss the chicken and secure it onto the rotisserie spit.
Cook: Cook for 1 to 1.5 hours or until the bird measures 165°F with a Thermapen in the breast meat.
Make the Sauce: While the chicken cooks, mix all green sauce ingredients in a food processor until smooth. Place in the refrigerator to allow flavors to meld.
Serve: Carve the chicken and serve with the green sauce.