Place eggs in a saucepan and cover with cold water by about 1 inch. Bring to a gentle boil over medium heat.

Once boiling, reduce heat to low and simmer for exactly 8 minutes for a creamy yet fully set yolk.

Immediately transfer eggs to a bowl of ice water and chill for at least 5 minutes—this helps ensure smooth peeling.

Peel the eggs and place into a bowl. Mash gently with a fork until slightly chunky.

Add mayonnaise, Dijon mustard (if using), sugar, salt, and pepper. Mix until creamy and well combined.

Lightly spread softened butter (if desired) on one side of each bread slice.

Divide egg salad evenly between two bread slices and top with the remaining two slices, butter-side down.

Gently press the sandwiches together, wrap in parchment paper or plastic wrap, and refrigerate for 10 minutes to let flavors meld and the sandwich hold shape.

Slice off the crusts (if any remain) and cut each sandwich in half on the diagonal. Serve chilled or at room temperature.