Place the eggs in a saucepan and cover with cold water.
Bring the water to a boil, then reduce to a gentle simmer and cook for 10 minutes.
Transfer the eggs to an ice bath and let them cool completely before peeling.
Separate two of the yolks from the whites.
Mash the two yolks into a smooth paste with the mayonnaise, mustard, sugar, milk, salt, and pepper.
Roughly chop the remaining eggs, including the extra yolks, and gently fold them into the creamy mixture.
Stir in the chopped chives.
Lightly butter one side of each slice of bread.
Spread a generous amount of egg salad onto four slices.
Top with the remaining bread slices, buttered side facing the filling.
Trim the crusts if desired.
Cut each sandwich in half or into triangles and serve immediately.