Heat oil in a large skillet over medium-high heat. Add onion and cook until softened, 3–4 minutes.
Add minced chicken, breaking it up with a spatula; cook until no longer pink, about 6–8 minutes.
Stir in garlic and ginger; cook 1 minute until fragrant.
Mix hoisin, soy sauce, and rice vinegar in a small bowl; pour into skillet and stir to coat.
Add shredded carrot and cook 2 minutes to soften. Taste and add red pepper flakes if desired.
Remove from heat and stir in green onions.
Spoon filling into lettuce leaves, sprinkle sesame seeds, and serve immediately.