Place the diced mango into a small saucepan. Add water and heat over low heat, stirring to prevent burning, for about 5 minutes.
Remove mango mixture from heat and mash with the back of a spoon or use a muddler. Set the mixture aside to cool slightly.
Place a strainer over a 2-cup glass measuring cup, and transfer the cooked mango to the strainer. Using a wooden spoon, press the juices from the mango into the cup.
Continue pressing until no more juice is left (you'll be left with pulp in the strainer).
The liquid will serve as the mango simple syrup for the cocktail (no sugar added!). At this point, if you have some fresh basil in your garden, muddle in a few leaves for even more deliciousness.