In a bowl, stir together the peanut butter, hoisin sauce and lime juice with half a cup of hot water.
Cook the noodles in boiling water according to package directions. Drain and set aside.
Lightly season the beef with salt and pepper.
Heat the cooking oil to smoking point. Stir fry the beef just until no longer pink. Remove with a slotted spoon and keep warm.
Reheat the oil. Stir fry the green beans and carrot for about a minute. Remove with a slotted spoon and set aside.
Reheat the oil once more. Saute the garlic, shallots and sliced chili until fragrant, about 2 minutes. Return the beef and vegetables to the pan. Season with fish sauce and more pepper. Pour in the peanut sauce and stir.
Add the cooked noodles and toss until coated with the sauce and heated through. Off the heat, drizzle with sesame seed oil and toss a few times before serving.