Cook noodles according to package directions, omitting salt and fat. Set noodles aside; keep warm.
Heat a large nonstick skillet over medium-high heat. Add 2 teaspoons peanut oil to pan; swirl to coat. Add carrot to pan; sauté 2 minutes. Add bell peppers; sauté 2 minutes. Remove carrot mixture from pan.
Heat the remaining 2 teaspoons oil in pan over medium-high heat; swirl to coat. Add mushrooms; sauté for 2 minutes. Add lemongrass and the next 5 ingredients (through garlic); cook for 1 minute, stirring constantly. Add broth, 1/2 cup water, soy sauce, and salt. Bring to a boil; cover, reduce heat, and simmer for 2 minutes or until slightly thick. Add noodles, carrot mixture, and onions; cook for 2 minutes, tossing to combine. Divide noodle mixture evenly among 4 bowls; top with cilantro and cashews.