Soak the Noodles: Place the noodles in a large heatproof bowl and cover with boiling water. Let them stand for 5 minutes, or until just tender, then drain and rinse well under cold water. Drain again thoroughly excess water will make the stir-fry steam instead of fry.
Make the Sauce: In a small jug, combine the fish sauce, brown sugar, and lemon juice. Stir until the sugar dissolves and set aside.
Cook the Chicken: Toss the chicken with half the peanut oil. Heat a wok over high heat, then add a third of the chicken at a time, stir-frying for 2 minutes or until just cooked through. Remove to a clean bowl and repeat in two more batches, adding a little more oil if needed.
Soften the Vegetables: Reduce the heat to medium and add the remaining oil along with the sliced onion. Stir for 3 minutes, or until softened. Add 2 teaspoons of water, cover, and cook for a further 2 minutes.
Build the Stir-Fry: Remove the lid and stir in the pad thai paste, stir-frying for 30 seconds until fragrant. Add the carrot and snow peas, stir-frying for 3 minutes, or until the carrot is just tender.
Scramble the Egg: Push the vegetables to one side of the wok. Pour the beaten egg into the cleared space and let it sit for a moment before stirring gently with a wooden spoon until scrambled.
Bring It Together: Increase the heat to high. Add the drained noodles, cooked chicken, and the reserved sauce. Stir-fry for 1 minute, or until everything is hot and well combined.
Serve: Remove from the heat and sprinkle with chopped peanuts. Serve immediately with bean sprouts, sliced chili, coriander leaves, and lemon wedges alongside for everyone to add as they like.