Combine the lemon juice and water in a large resealable plastic bag and place in the kitchen sink or a large bowl. Add the chicken to the acidulated water and let sit while you prepare the marinade.
Make the marinade: In a blender jar or the bowl of a mini food processor, combine the lime juice, garlic, aji amarillo and huacatay pastes, if using, the oil, paprika, cumin, sugar, salt, and pepper. Process until relatively smooth.
Drain the chicken and pat dry. Return the chicken to the resealable plastic bag. Pour the marinade over the chicken and massage the bag to evenly coat the bird. Refrigerate for at least 8 hours, and up to 24.
Up to 2 hours before you cook the chicken, make the Green Sauce. Or make it while the chicken roasts: In a blender jar or the bowl of a mini food processor, combine all the sauce ingredients and process until smooth. Taste for seasoning, adding more salt and pepper if needed. The sauce should be very flavorful. Cover and refrigerate if not using immediately.
When ready to cook, start the Traeger grill on Smoke with the lid open until the fire is established (4 to 5 minutes). Set the temperature to 375F and preheat, lid closed, for 10 to 15 minutes.
Remove the chicken from the marinade and discard the marinade. Tuck the wings behind the back and tie the legs together with butcher's string. Place the bird directly on the grill grate. Grill until the skin is crisp and brown and the internal temperature of the thigh when read on an instant-read meat thermometer is 165F, about 70 to 80 minutes.
Transfer the chicken to a platter or cutting board. Let the chicken rest for 10 minutes before carving. Serve the Green Sauce on the side.