Bring a pot of water to a rolling boil and carefully add the 7 eggs, then immediately start timing. For soft-boiled eggs with a slightly creamy yolk (the traditional tamago sando style), cook for exactly 7 minutes, then remove from heat and let sit undisturbed for 1 minute. While the eggs cook, prepare a bowl filled with ice and cold water—this will stop the cooking process immediately and ensure the yolks stay perfectly creamy. Once the time is up, transfer the eggs to the ice bath and let them cool for 2 minutes, then gently peel under cool running water, taking care to preserve the tender whites. In a bowl, combine the peeled eggs, sugar, salt, and freshly ground pepper, then mash with a fork until you reach your desired consistency—I prefer leaving some small chunks for texture rather than making it completely smooth.
To the mashed egg mixture from Step 1, fold in the mayonnaise and milk using gentle stirring motions to keep the texture slightly chunky rather than completely smooth. Add the pinch of Dijon mustard and stir to combine—the mustard adds a subtle depth without being noticeable, which is exactly what you want in a tamago sando. Taste and adjust the seasonings if needed, keeping in mind that the butter and bread will add mild flavors. Let the filling sit for a moment while you prepare the bread; I find the flavors meld together beautifully in just a couple of minutes.
Lay out all 4 slices of milk bread on a clean surface. Using a butter knife, spread the room-temperature butter evenly across one side of each slice—room temperature butter spreads without tearing the delicate bread. Divide the egg filling from Step 2 between two slices of bread, spreading it gently and evenly to the edges, then top each with the remaining buttered slices, butter-side down, to create two sandwiches. Press down gently on the top to help the ingredients bond with the soft bread.
Using a sharp, dry knife (wipe it clean between cuts), carefully remove the crusts from all sides of each sandwich, cutting away about 1/4 inch of crust to reveal the pristine interior. Cut each sandwich diagonally into two triangles—this is the classic tamago sando presentation and makes the sandwich easier to eat. Arrange the triangles on a serving plate and garnish with fresh chives for a pop of color and a subtle onion note that complements the richness of the egg salad.