Heat olive oil in a saucepan over medium-high heat.
Add ground chicken and cook until browned, about 3-5 minutes, making sure to crumble the chicken as it cooks; drain excess fat.
Stir in garlic, onion, hoisin sauce, soy sauce, rice wine vinegar, ginger, and Sriracha until onions become translucent, about 1-2 minutes.
Stir in chestnuts and green onions until tender, about 1-2 minutes; season with salt and pepper, to taste.
To serve, spoon several tablespoons of the chicken mixture into the center of a lettuce leaf.