In a mini chopper or food processor, combine basil, cilantro, lime juice, lemongrass, ginger, oil, garlic, salt and pepper; pulse until smooth. Add water, 1 teaspoon at a time, to thin sauce (if desired). Spoon sauce in a small bowl and add carrots; toss to combine and set aside.
Coat a grill or grill pan with cooking spray; preheat to medium-high.
Pound chicken until 1/4-inch thick (to see a video about pounding chicken, click here). Season both sides of chicken with pepper, if desired (you don’t need salt because the sauce is very flavorful). Place chicken on grill and cook until browned and cooked through, flipping once, about 2 to 3 minutes per side.
Place 1 piece of chicken on bottom half of a roll; top with 1/4 cup carrot mixture, 1/4 of cucumber slices and a roll top. Repeat with remaining ingredients and serve. Yields 1 sandwich per serving.