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. Fold in the cilantro. Set aside.
Heat the oil in a large skillet over medium-high heat. Pat the chicken dry with paper towels. Season with ½ teaspoon salt and ¼ teaspoon pepper.
Cook the chicken until golden brown and cooked through, 2 to 3 minutes per side. Transfer to plates.
In a bowl, combine the chili sauce, soy sauce, and remaining 1 tablespoon vinegar, 2 teaspoons lime juice, and 2 teaspoons brown sugar.
Stir in the peanut butter and ¼ cup hot water. Spoon the sauce into small ramekins or bowls for dipping.
Stir the cucumber salad and serve it with the chicken and dipping sauce.