Heat sesame oil in a large skillet over medium-high heat. Add white parts of green onions, garlic, and ginger; cook about 1 minute until fragrant.
Add mushrooms and cook 3-4 minutes until they release moisture and start to brown.
Add ground turkey and cook, breaking it up, until no pink remains.
Whisk together soy sauce, rice vinegar, honey, and sriracha (if using). Pour over the turkey mixture and cook 1-2 minutes until the sauce coats everything.
Spoon into lettuce leaves and top with shredded carrot, green parts of the green onions, and sesame seeds.