Warm sesame oil and 1 Tbsp. vegetable oil in a skillet over medium-high heat. Add onion and cook, stirring, until softened, 3 minutes. Stir in mushrooms, chicken and rice mix. Pour in broth and increase heat to high. Bring to a boil, then reduce heat to medium-low. Cover and cook until rice is tender and liquid is absorbed, 25 minutes. Transfer to a bowl; let cool.
In a large bowl, whisk vinegar, salt and mustard until salt is dissolved. Slowly whisk in remaining vegetable oil until blended.
Bring a large pot of salted water to a boil; add snow peas. Bring back to a boil; cook until peas are tender, 1 minute. Drain; rinse under cold water. Pat dry.
Add chicken and rice mixture to bowl with dressing. Toss with snow peas, carrots and water chestnuts. Serve at room temperature.