Place chicken in a small bowl and cover with 1 tablespoon of fish sauce; set aside.
Steam broccoli and carrot until crisp-tender, about 8 to 10 minutes. Immediately immerse in cold water; drain and set aside.
Meanwhile, in another small bowl, combine remaining 3 tablespoons of fish sauce, 3 tablespoons of lime juice, sugar, water and ginger; mix well and set aside.
Coat a large wok or skillet with cooking spray and warm over medium-high heat; add 2 teaspoons of oil. When hot, add chicken; cook, stirring occasionally until chicken is cooked through, about 4 minutes. Remove chicken from pan and cover to keep warm; set aside.
Drain noodles thoroughly. Off heat, coat same wok or skillet with cooking spray and warm over medium-high heat; add remaining 2 teaspoons of oil. When hot, sauté garlic and chili pepper until just fragrant, about 30 seconds. Add noodles and fish sauce mixture; cook, stirring, until sauce is mostly absorbed, about 3 minutes. Stir in chicken and peanuts; remove from heat. Add remaining tablespoon of lime juice, bean sprouts and cilantro; mix well. Serve immediately. Yields about 2 cups per serving.