Peel the mangoes, and then cut them into cubes about 3/4 inch thick and put them in a bowl.Squeeze the juice from any pulp remaining on the pit into a separate small bowl and set aside.
Drain the lychees, reserving 1 tablespoon of the syrup.
Put the brown sugar and the tablespoon of lychee syrup in a heavy saucepan over high heat. Cook and stir for 2 - 3 minutes, until the sugar dissolves and the mixture caramelizes. Add the mangoes and stir gently a few seconds. Then add the lychees, reserved mango juice, lime juice, and ginger. Shake the pan as the mixture bubbles for about 1 minute. Remove from heat and let cool slightly.
Gather the mint leaves into 2 to 3 stacks. Slice them crosswise into 1/8-inch shreds. Fluff slightly to separate them.
Divide the mango mixture among 4 bowls. Scatter one-quarter of the mint leaves on top of each serving. Serve at once.
Optional: top with your favorite sorbet!