Heat the oil in a small saucepan over medium heat. Add the garlic and cook for 2 to 4 minutes or until fragrant and pale golden brown. Be careful not to let the garlic burn.
Remove the pan from the heat and stir in the bitter orange marinade (or lime and orange juice combo), vegetable broth or stock, cumin, oregano and pepper.
Return the pan to the stove and bring the sauce to a boil over high heat. Once it boils, reduce the heat to medium and let simmer for 3 minutes or until the flavor has mellowed. Season with salt. Remove from the heat and let cool to room temperature.
Stir in the cilantro and heat again right before serving. Store in a glass jar or bottle in the refrigerator for up more than a month. Shake well and reheat leftover sauce before using.
In a baking dish, combine the bitter orange marinade, olive oil, salt, pepper and paprika and stir thoroughly. Place the chicken halves in the marinade and cover with plastic wrap and allow to sit preferably overnight. Turn the chicken once in a while to evenly coat the chicken with the marinade.
Light a charcoal or wood fire or fire up a gas grill. Place chicken directly over the coals or flame for 5 minutes or until a slight char is obtained on the skin. Flip and cook for another 5 minutes on the other side.
Move chicken to indirect heat and close the lid. Cook for 40-50 minutes until the internal temperature reaches 180-190 degrees. During the cooking process, baste the chicken every 10 minutes with the prepared mojo sauce.
Towards the end of the cooking process, move the chicken over the direct heat for additional crisping. When cooking temperature has been achieved, remove the chickens from the grill and let rest for a few minutes. You may either serve it whole or cut the half into pieces.
Pour mojo sauce liberally over the chicken and serve.