Preheat the oven to 375 degrees F. Place the chicken in a parchment paper-lined baking dish. Combine the sweet chili sauce and 2 Tbsp of the lime juice in a small bowl; pour over the chicken. Bake 20 to 25 minutes, basting occasionally, or until cooked through. Cool the chicken to room temperature; cover and chill in the fridge at least 2 hours.
Meanwhile, cook the noodles in boiling water for 1 minute; drain well. Submerge the noodles in cold water to cool, and then drain well again. Whisk the vegetable oil, sesame oil, soy sauce, sugar, garlic, ginger and remaining lime juice in a medium bowl. Add the noodles the remaining ingredients, except garnish, and gently toss to combine. Divide the salad among plates. Slice the chicken and arrange on top, drizzling it with any glaze remaining in the baking dish. Garnish with cilantro springs and lime wedges and serve.
Note: Thai-style sweet chili sauce and Chinese-style egg noodles are available in the Asian foods aisle.