The fish sauce is very salty, so season sparingly.
Heat a 12-inch nonstick skillet over medium heat. Add the turkey and broth; cook until the turkey is no longer pink, about 8 minutes. Remove with a slotted spoon; set aside. To the remaining broth in the pan, add the shallots, scallions, fish sauce, lemongrass, and chiles. Mix well and cook 1 minute. Remove from heat. Stir in the turkey, mint leaves, and lime juice. Check seasoning; add more salt if needed. Refrigerate at least 1 hour or overnight. Bring the turkey mixture to room temperature before serving.
To serve, place about ¼ cup turkey mixture in the center of each lettuce leaf. Leave unwrapped. Arrange the leaves on a platter and serve the wraps with bowls of mint leaves, lime wedges, and cucumber slivers.
To eat, take a filled leaf and squeeze some lime juice over the turkey mixture. Sprinkle mint and cucumber slivers over mixture. Fold in the left and right edges of the lettuce; roll into a cylinder. Garnish with a wedge of lime.