Remove backbone of chicken with poultry shears; flatten chicken out by pressing firmly on skin part of chicken so it will sit evenly in a pan (How to Spatchcock a Chicken
Preheat oven to 300 degrees F.
Combine all the dry spices and rub chicken with spice mixture, especially under the skin. Put into a resealable plastic bag and refrigerate for 30 minutes.
In a large mixing bowl combine all the ingredients for the marinade, then add chicken to marinade and refrigerate for at least 1 hour.
Remove chicken from marinade and shake off excess marinade. In a large saute pan over medium to high heat, add 4 tablespoons of extra- virgin olive oil. When oil is hot, place chicken skin side down and sear.
When skin is golden brown flip chicken and add 1/2 cup of marinade into pan; place pan in oven and cook the chicken for 25 minutes or until chicken has internal temperature of 165 degrees F. on an instant-read thermometer.
Glaze: In medium sauce pot, place rum, chicken broth and brown sugar. Reduce by 1/3 over high heat. In a small mixing bowl, whisk together water and cornstarch. When rum mixture is reduced, add cornstarch mixture slowly to simmering liquid and whisk for 3 minutes, until 50 percent thicker. Keep in mind that you might not need all of it. When glaze is at desired thickness, add mint leaves and transfer to small bowl. Season with salt and pepper, to taste.
Remove the chicken from oven and brush with Mojito Glaze. Place chicken under broiler for 5 minutes. Remove from broiler, cut and serve immediately with more glaze on the side.