In a microwave-safe bowl, whisk soy sauce and brown sugar together until dissolved then add gradually to peanut butter; stir until smooth; stir in ginger, sesame oil, chili oil, and rice vinegar; blend well.
FYI: At this point the thick peanut sauce makes a great dipping sauce!
Stir chicken broth into peanut sauce; cook in microwave oven about 45 seconds or just until bubbly around edges; stir and set aside.
Bring a large saucepan containing 5-6 cups water to a boil; remove pan from heat and add rice noodles; soak noodles in hot water for 3-5 minutes until they are cooked but still al denté firm; rinse noodles under cold water; drain and set aside; if your noodles are very long you may want to chop them into a more manageable length.
Meanwhile, peel carrots then slice them with a diagonal cut into pieces about 1/8" thick and 2" long; par boil or steam carrots slices together with frozen broccoli cuts for about 3 minutes or until carrots are barely crisp-tender and broccoli is no longer frozen; remove from heat and rinse under cold water; trim down any oversized pieces of broccoli; set aside.
Add peanut oil and sesame oil to a large nonstick skillet or wok over medium-high heat; heat until oil is almost smoking; add chicken to skillet and sauté 1 1/2 minutes or until the chicken is not quite done; add mushrooms, carrots, and broccoli to skillet; sauté another 1 1/2 minutes; add peanut sauce and toss about 30 seconds until chicken and veggies are evenly coated with sauce; add rice noodles; cook and toss for 1 minute until heated through.
Turn onto serving plate(s) and serve immediately garnished with sliced green onions and chopped peanuts.