In a medium pot filled with enough water, bring to a boil over high heat. Carefully lower eggs into boiling water. Reduce to medium-high heat and boil eggs for 13 minutes covered. We need these to be hard-boiled in order to easily separate the egg whites from yolks. Tip: add salt and vinegar to prevent any cracked eggs from leaking out and the vinegar helps make the egg peeling process so much easier!
Once boiled, remove hard-boiled eggs with a slotted spoon and transfer them to a large bowl filled with cold water and ice (or an ice bath).
Once eggs are cool to touch, crack and peel them with the back of the spoon. Tip: Start from the less pointy side of the egg as there's a small gap where you can insert the spoon to easily peel away the egg shell. At this point, you can either follow Steps 4-6 for the traditional aesthetic method OR skip steps 4-6 and mash everything together with a fork.
Carefully run a sharp knife around the yolk to gently separate egg yolks from the whites. Transfer whites into another large bowl.
Then run yolks through a fine sieve set over a large mixing bowl. Use the back of a spatula or spoon to push the egg through the sieve until your egg yolks are finely grated. Make sure to scrape the back of the sieve for more egg yolk.
Finely dice egg whites with a knife. Transfer egg whites into bowl with egg yolk.
To your eggs, add salt, black pepper, sugar and Japanese mayo. Mix with a spatula until creamy.
Cut your brioche buns into half about 95% of the way. If you're using sandwich bread, skip this step.
Scoop an equal portion of your egg salad onto the bun or a slice of bread and close it with the other piece. Optional: Garnish with dried parsley. Enjoy!