In a bowl, whisk together the lime juice, rum, chopped mint, mint syrup and vegetable oil.
Place the chicken in large plastic zip bag. Pour the marinade over and seal the bag. Turn the bag to cover. Chill at least 4 hours, turning several times during the day to re-distribute the marinade.
GRILL THE CHICKEN:
Place the lime quarters in a shallow bowl. Pour 2 tablespoons of rum over, tossing to coat. Let stand at room temperature.
Prepare the grill to medium heat. Remove the chicken breast from the marinade and grill until cooked through, about 10 minutes per side (time will vary based on thickness of chicken). Transfer to a platter.
Grill the limes until soft and slightly charred, about 5 minutes. Garnish the platter with mint sprigs. Squeeze the grilled limes over the chicken and serve.
Notes
Mint syrup is often shelved with the coffee and tea at the supermarket. It's available anywhere that sells syrups for coffee. To make your own: Mix 1 cup of sugar and 1 cup of water and boil for one minute. Remove from heat and add a handful of mint leaves. Let sit until cooled. Strain out the mint and you'll have mint syrup remaining. Measure from that.
The longer you marinate the chicken, the more flavorful it will be.
You can use boneless chicken breasts for this, but they're just not as good as the bone-in for this recipe.
To make grilling easier, you can bake the breasts for a short time and then finish them off on the grill. Bake at 350 for 25 minutes (covered) and then finish off on the grill about 5 min. per side.