Combine first 6 ingredients in a medium saucepan. Bring mixture to a boil; cook 20 minutes. Remove from heat. Stir in carrot; let stand 10 minutes to soften. Add chicken to carrot mixture; let stand 5 minutes. Drain and discard liquid.
Place about 1/4 cup chicken mixture in each lettuce leaf, and top each lettuce leaf with 2 cucumber slices, 1 1/2 teaspoons peanuts, and 1 teaspoon mint.
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