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Add enough water to a deep cooking pan to cover the eggs. Bring the water to a gentle boil over medium-high heat. When tiny bubbles appear on the sides of the pan, carefully add the eggs.
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Cook the eggs for 6 minutes for runny yolks. (Note: I used large eggs. Results may vary depending on the size of the eggs.) Cook the eggs for up to 10 minutes if you prefer medium or hard-boiled eggs.
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Meanwhile, prepare an ice water bath. When the cooking time is up, transfer the eggs to the bowl of ice water and let them cool.
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For the sauce, mix all the ingredients listed under the marinade.
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Once the eggs are completely cooled, peel them very carefully. Transfer them to a container with a lid, pour the marinade over the eggs, cover, and refrigerate for at least six hours.
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Serve the eggs as a topping over warm rice, and drizzle with sesame oil.