In a medium bowl, stir vinegar, sugar, and 1 teaspoon salt until sugar is dissolved. Add carrot and mango; toss to combine. Let stand at room temperature, tossing occasionally, at least 30 minutes, or refrigerate up to a day.
Heat grill or grill pan to high; lightly oil grates. In a shallow dish, combine lime juice, ginger, and oil; season with salt and pepper. Add chicken and let marinate for no more than 5 minutes. Grill chicken until browned and opaque throughout, 1 to 3 minutes per side; transfer to a plate to cool.
Thinly slice chicken; toss with carrot mixture. Divide mixture among lettuce leaves; drizzle with juice from carrots and sprinkle with cilantro.
If you canÂ‚'t locate a green (unripe) mango, pick a firm one. To store, refrigerate carrot and mango mixture, grilled chicken (unsliced), and cleaned lettuce leaves, separately, up to a day.