Heat 1 Tablespoon olive oil in a large skillet over medium heat. Add the ground chicken and cook, crumbling, for 2-3 minutes. (Chicken will not be fully cooked.)
To the same skillet, add the chopped mushrooms, water chestnuts, carrots, ginger, and garlic. Cook for an additional 2-3 minutes, until chicken is fully cooked and vegetables are tender-crisp.
In a small bowl, combine hoisin sauce, soy sauce, chili garlic sauce, and sesame oil to make the sauce. Mix well.
Pour the sauce over the chicken & vegetables. Cook, stirring, over medium heat for 1-2 minutes until sauce is incorporated.
Spoon filling into lettuce cups; garnish with sliced green onions and toasted sesame seeds. Drizzle with additional sauce (or sauce of your choice) as desired.