Combine the chicken, lime juice, cilantro, garlic, sugar, soy sauce, pepper and mint in a plastic, re-sealable food storage bag and seal. Shake to combine, then transfer chicken to the refrigerator and allow to marinate for 24 hours. Remove the chicken from the refrigerator and discard marinade. Allow chicken to come to room temperature for 20 minutes.
Preheat a grill to high.
When the grill is hot, grill the chicken breasts, turning to mark on both sides, until an internal temperature of 165 degrees F, about 8 to 10 minutes, depending on your grill. Remove from the grill and allow to rest briefly before carving each portion into thin slices.
When ready to serve, place 2 lettuce leaves on each of 4 plates, and fan the chicken slices on top of the lettuce leaves. Garnish each plate with 1 1/2 ounces of the mango, 1 1/2 ounces of the papaya, 1 1/2 ounces of the pineapple and 5 marcona almonds. Drizzle 2 tablespoons of the Lemon-Ginger Vinaigrette over each salad and serve immediately.
Combine all ingredients in a nonreactive container with a tight fitting lid, cover, and shake to combine. Serve immediately or refrigerate for up to 2 days before serving.