Heat oil in wok over medium heat. When oil reaches 350 degrees add the sliced ginger and fry until lightly brown and crispy, about 4 minutes. Remove to a plate lined with a paper towel and reserve as garnish. Carefully pour out half the oil from the wok into a heatproof container and reserve for frying egg.
Return wok to the stove and turn the heat up to high; add onion, carrot, ginger sticks and tofu. Stir-fry for about 2 minutes. Stir in chayote, fish sauce, sweet soy sauce and regular soy sauce and continue to cook for another 2 minutes. Add scallions, lime juice and egg noodles and stir. Cook until noodles are heated through and transfer to large serving bowl.
Add reserved oil to wok and heat over medium-high heat. Crack egg into a small bowl and carefully slide the egg into the hot oil. Fry for 3 minutes until egg white is cooked but yolk is still runny. Place egg on top of stir fry and garnish with scallions and fried ginger chips.