Bring a pot of water to a full rolling boil. Prepare a large bowl with cold water and ice. Carefully lower all 7 room-temperature eggs into the boiling water and start a timer.
At exactly 7 minutes and 30 seconds, remove 1 egg and drop it into the ice bath. At the 9-minute mark, remove the remaining 6 eggs and place them in the ice bath. Let cool for 2 minutes, then peel gently.
Slice the 6 hard-boiled eggs in half, transfer to a mixing bowl, and mash with a fork until finely minced.
Add the milk, Kewpie mayonnaise, sugar, salt, black pepper, and chives. Mix until creamy and glossy. Chill in the fridge for 10 minutes.
Spread a thin, even layer of softened butter on one side of all 4 bread slices to prevent the sandwich from getting soggy.
Halve the medium-boiled egg lengthwise. Place one half, cut-side down, in the exact center of two bread slices.
Spoon the chilled egg salad around the medium-boiled egg halves, spreading evenly to the edges. Place the remaining bread slices on top, butter-side down. Press lightly.
Wipe a sharp knife with a damp cloth, slice each sandwich straight down the middle to reveal the center egg, and serve immediately.