Heat vegetable oil in a large skillet or wok over medium-high heat. Add the ground chicken and cook for 6 to 7 minutes, breaking it into small crumbles, until browned and cooked through.
Add the minced garlic and grated ginger. Stir-fry for 30 seconds until fragrant and the garlic is just starting to turn golden.
In a small bowl, stir together the soy sauce, hoisin sauce, rice vinegar, and sriracha. Pour the sauce mixture over the chicken and stir to combine.
Add the diced water chestnuts and most of the sliced green onions. Toss everything together and cook for 2 more minutes so the water chestnuts warm through and the sauce reduces slightly.
Drizzle the sesame oil over the mixture and give it one final toss.
Spoon the chicken mixture into butter lettuce cups. Top with remaining green onions, shredded carrots, and a sprinkle of sesame seeds. Serve immediately.