Make Peanut Sauce: In a small bowl, whisk together peanut butter, soy sauce, lime juice, brown sugar, and ginger. Add 2-3 tablespoons of water, if needed, until the desired smooth consistency is reached. Set the sauce in the fridge until ready to use, and re-whisk just before serving.
Prep The Vegetables: Peel and shred the carrots, finely dice the shallots, mince the garlic, grate the ginger and chop the cilantro.
Cook The Chicken Mixture: Heat vegetable oil and sesame oil in a large skillet over medium-high heat. Add the ground chicken, carrots, and shallots, crumbling the chicken as it cooks.
Stir in Garlic and Ginger: Once the chicken is almost browned, stir in the garlic and ginger, and cook for a few more minutes until the chicken is browned.
Stir In Hoisin Sauce: Stir in the hoisin sauce and cook until heated through, about 1 minute.
Finish With Cilantro: Remove the skillet from heat and stir in the chopped cilantro.
To Serve: Spoon several tablespoons of the chicken mixture into the center of a lettuce leaf, taco-style. Drizzle with peanut sauce before serving.