Heat a wok or large frying pan over high heat. Add oil and swirl to coat pan. Cook chicken, stirring only when browned, until most of pink is gone, about 10 minutes. Add ginger, garlic, onion, and carrot, stirring to combine.
Make a well in the center of the pan and pour in broth. Cook until simmering. Mix cornstarch and soy sauce in a small bowl. Add chili sauce, then pour into broth, stirring to combine. Add Little Gems and toss mixture to coat, cooking until half the lettuce leaves are wilted, about 10 minutes.
Sprinkle the dish with basil.
*A mini romaine, Little Gem lettuce has sweet, crunchy, and densely packed leaves. Find it at farmers' markets and well-stocked grocery stores.