Make the marinade by whisking together 2 Tbsp olive oil, 3 Tbsp soy sauce, 2 pressed garlic cloves, 1 tsp grated fresh ginger (or 1/4 tsp dried), and 1/2 Tbsp sesame oil in a small bowl.
Place the chicken breasts in a bowl, pour the marinade over them, and toss to coat. Refrigerate for 20 minutes while you prepare the vegetables and other components.
Heat a large skillet over medium heat and add the chicken along with any remaining marinade. Sauté until browned and cooked through, 6–10 minutes per side depending on thickness; cover briefly if needed to finish cooking.
Transfer the chicken to a cutting board and let rest 5 minutes, then slice thinly on the diagonal into strips.
Rinse 3/4 cup dry quinoa in a fine-mesh colander under cold water until water runs clear. Cook with 1 1/2 cups filtered water in a rice cooker on white rice setting or simmer on the stove until tender (about 15–20 minutes). Fluff and let cool to room temperature.
Make the peanut sauce by combining 3/4 cup sesame ginger dressing with 2 Tbsp creamy peanut butter in a small bowl and whisking until smooth. Top with chopped peanuts if using.
Arrange the lettuce leaves, sliced cucumber, carrot matchsticks, cilantro sprigs, sliced chicken, and cooked quinoa on a platter for serving.
To assemble, place a heaping tablespoon of quinoa on a lettuce leaf, add one or two chicken strips, cucumber and carrot, a few cilantro leaves, and drizzle with peanut sauce. Serve immediately.