Heat vegetable oil and sesame oil in a large cast iron skillet over medium high heat. Add ground chicken, mushrooms, carrot and shallot. Cook until browned, about 5-7 minutes, making sure to crumble the chicken as it cooks; drain excess fat.
Stir in garlic and ginger until fragrant, about 1 minute.
Stir in hoisin sauce, chili garlic sauce and fish sauce until heated through, about 30 seconds. Remove from heat; stir in cilantro.
To serve, spoon several tablespoons of the chicken mixture into the center of a lettuce leaf, taco-style, drizzled with peanut sauce.
For the peanut sauce
In a small bowl, whisk together peanut butter, soy sauce, lime juice, brown sugar, chili garlic sauce and ginger. Whisk in 2-3 tablespoons water until desired consistency is reached.