Bring egg-cooking water to a gentle boil in a saucepan. Lower in eggs carefully and simmer for 11 minutes, then transfer to the ice and cold bath water for 10 minutes.
Peel eggs and separate the yolks from the whites. Press yolks through a fine sieve into a bowl and stir with Japanese mayonnaise, milk, salt and white pepper until smooth.
Chop whites finely and fold into the yolk mixture. Cover and refrigerate for 30 minutes until thoroughly cold; chilling is included in prep time.
Spread butter thinly over one face of each bread slice. Divide egg salad over 2 slices, smoothing it into an even layer almost to the edges.
Close with the remaining bread, butter-side down. Trim the crusts carefully, cut each sandwich diagonally and serve cold or refrigerate promptly at 40°F or below.