Combine 3 tablespoons of the vinegar, 2 teaspoons of the lime juice, 1 teaspoon of the brown sugar, and the cucumbers in a medium bowl. Gently fold in the cilantro and set aside.
Heat the oil in a large skillet over medium-high heat.
Pat the chicken dry with paper towels and season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook until golden brown and cooked through, 2 to 3 minutes per side. Transfer to individual plates.
Combine the chili sauce, soy sauce, and the remaining vinegar, lime juice, and brown sugar in a bowl. Stir in the peanut butter and 1/4 cup hot water. Spoon into small ramekins or bowls for dipping.
Stir the cucumber salad and serve it with the chicken and dipping sauce.
Shortcut: Save time by using a prepared peanut dipping sauce (sometimes called satay sauce) in place of the peanut-butter and soy-sauce mixture in the recipe below. You can find it in the international aisle at most supermarkets.