In a large bowl, combine chili sauce, soy sauce, lime juice, oil, Sriracha, ginger, and garlic. Whisk to combine. Add chicken breasts to bowl and toss with marinade.
Heat a large grill pan over medium heat. Add chicken breasts and cook until golden with grill marks and no pink remaining, 8 to 10 minutes per side. Transfer to a plate to let rest, then thinly slice.
Build your lettuce cups: Fill lettuce cups with chicken and garnish with onion, carrot, and cilantro before serving.