Mix lemongrass, 1 1/2 tablespoon vegetable oil, lemon juice, soy sauce, brown sugar, and garlic together in a bowl until sugar is dissolved; add chicken and turn to coat in the marinade.
Marinate chicken in the refrigerator for 20 minutes to 1 hour.
Preheat grill for medium heat and lightly oil the grate. Transfer chicken to the hot grill, discarding marinade. Grill chicken until no longer pink in the center, about 6 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Whisk fish sauce, 2 tablespoons vegetable oil, rice vinegar, white sugar, water, garlic powder, and red pepper flakes together in a bowl to make dressing.
Place noodles in a large bowl and cover with hot water. Let soak until softened, about 15 minutes. Drain and rinse thoroughly.
Combine noodles, lettuce, bean sprouts, carrot, cucumber, and mint leaves in a large bowl. Slice grilled chicken and arrange over salad; drizzle dressing on top.