Prepare the rice noodles according to package instructions until slightly al dente; rinse in cold water, drain, toss with 1 teaspoon toasted sesame oil, and set aside.
Whisk together the sauce: brown sugar, soy sauce, fish sauce, 3 tablespoons lime juice, rice vinegar, peanut butter, chili paste, dried basil, ground coriander, ground ginger, and pepper; set aside.
Heat 1 tablespoon vegetable or peanut oil in a large nonstick skillet or wok over medium-high heat; when very hot add the sliced chicken and cook undisturbed 1 minute, then stir-fry just until cooked through; transfer chicken to a plate and wipe out the pan.
Add the remaining 1 tablespoon oil to the same skillet and heat over medium-high; add bell pepper and shallots and sauté 1 minute, then add matchstick carrots and minced garlic and sauté 1 more minute.
Reduce heat to medium, pour the beaten eggs over the vegetables, and stir constantly until the eggs are just scrambled and cooked.
Increase heat to medium-high, add the sauce and noodles to the pan, and toss to combine; stir and cook 1–2 minutes until noodles are al dente and warmed through.
Stir in the cooked chicken, bean sprouts, chopped green onions, and chopped peanuts and heat through, adjusting chili paste or an extra tablespoon lime juice to taste if desired.
Remove from heat, sprinkle with chopped cilantro, and serve immediately.
