Trim and butterfly chicken breasts. Salt and pepper to taste.
At this point you can either grill the chicken breasts or cook in a medium skillet over medium-high with a small amount of olive oil. The chicken will need to cooked long enough that juices run clear. You'll prepare the mango mixture while the chicken cooks.
Squeeze juice from lime into small dish and combine with vinegars.
Finely chop cilantro and add to lime juice/vinegar mixture. Add the crystallized ginger and salt and set aside.
Peel and dice mango into a bowl. Set aside.
Remove the seeds from the jalapeno, and finally chop 1/2. Add to the mango. You can chop and use the remainder for a spicier dish.
In a small skillet, heat a small amount of olive oil over medium heat.
While the oil heats, mince garlic and then saute in oil for 1-2 minutes.
Add the mango/jalapeno mixture to the garlic/oil and heat for 3 minutes over medium.
Add the lime/vinegar/ginger to the skillet. Reduce heat to medium-low and continue to cook for 3 more minutes, stirring constantly. The mixture should thicken from the sugar in the mango. If the chicken is still not ready, keep the mixture warm over low heat.
Spoon the mango mixture over the cooked chicken breast and serve.