Bring 6 cups water, ½ teaspoon salt and 2 tablespoons of the distilled white vinegar to a boil in a medium pot over medium-high heat; prepare an ice bath. Carefully lower the eggs into the boiling water, then lower the heat to medium to keep it at a gentle, rolling boil. Let boil about 6 minutes for soft-boiled eggs. (If you want the yolks a little more set--cooked but still slightly jammy--boil for an additional 2 minutes.) Transfer immediately to the prepared ice bath and let sit until cool enough to handle, about 8 minutes.
Meanwhile, to make the marinade, whisk together the soy sauce, sugar, mirin, sesame seeds, garlic, Fresno chile, scallions, remaining 1 tablespoon distilled white vinegar and ½ cup water in an airtight container.
Carefully crack and peel the eggs, then drop them into the marinade. Marinate, refrigerated, for at least 6 hours and up overnight. Serve with warm white rice and drizzle with sesame oil.
