Combine the light soy sauce, vinegar, ginger, garlic and 1 tablespoon sugar in a jug. Whisk well to dissolve the sugar. Set aside.
Heat 2 teaspoons oil in a large wok until smoking. Add the half the chicken and stir-fry for 5 minutes or until golden. Use tongs to transfer the chicken to a plate. Repeat with the remaining chicken. Return the chicken to the pan along with half the soy sauce mixture and the remaining 2 tablespoons sugar. Cook, stirring often, for 8-10 minutes or until the sauce has reduced and the chicken is golden and sticky. Add the dark soy sauce and toss to combine. Sprinkle with half the sesame seeds.
Meanwhile, cook the noodles following the packet directions. Drain. Transfer to a bowl. Steam the snow peas until bright green and tender-crisp. Add the remaining 2 teaspoons oil to the remaining soy sauce mixture and stir to combine.
Divide the noodles among bowls. Top with the chicken. Drizzle with the remaining soy sauce mixture and sprinkle with remaining sesame seeds. Top with the snow peas and carrot. Scatter over the coriander and chilli. Serve with lime wedges, if you like.
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