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Place eggs in a medium-sized pot and cover with water. Bring to a boil over moderate heat, then turn off heat, cover pot and let eggs sit for about 4 minutes. Remove eggs from heat and transfer to a bowl of ice water to cool. Set aside.
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While eggs cook, add vegetable oil to a hot skillet and sear pork belly on all sides until a golden crust forms. Remove from skillet and slice thinly. Set aside.
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Heat a large soup pot over medium heat. Add sesame oil, garlic, ginger and chili garlic sauce and saute for 1-2 minutes. Add broth to pot along with soy sauce and rice vinegar. Bring to a simmer.
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Add noodles, mushrooms and the stalks of bok choy, reserving the leafy greens. Simmer gently for about 5 minutes. Taste and add more soy sauce if necessary.
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Divide soup evenly among four bowls. Carefully peel and halve the soft boiled eggs and place one egg each in a bowl. Add pork belly, bok choy leaves and scallions.