Place eggs in a saucepan and cover with cold water. Bring to a boil, then lower heat and simmer for 10 minutes.
Transfer eggs to ice water until cool enough to handle. Peel carefully.
Mash three eggs in a bowl with mayonnaise, sugar, and salt until creamy but with small chunks.
Slice the remaining egg in half. Set aside for presentation.
Spread a thin layer of butter on the bread slices.
Spoon the egg mixture onto one slice of bread, smooth it evenly, then top with the halved egg.
Close with the second slice of bread. Trim crusts if desired for a neater look.
Cut the sandwich in half to reveal the egg. Serve immediately.