Place the eggs in a single layer in a medium saucepan. Fill the pan with water until the eggs are just covered.
Transfer the saucepan to the stovetop, heating on medium-high until the water comes to a rolling boil.
Boil for 30 seconds, then immediately remove the pan from the heat, cover it, and set a timer for 11 minutes.
Meanwhile, fill one small and one large mixing bowl with cold water and ice.
Remove 1 egg from the hot water after 6 minutes have elapsed and place it in the small ice bath. This is the semi-soft boiled egg that will be reserved for decorating the sandwiches.
Remove the remaining 3 hard-boiled eggs after 11 minutes and place them in the larger ice bath.
After the eggs have cooled completely, about 10 minutes, remove them from the ice baths and peel them, making sure to keep the semi-soft boiled egg separate.
Place the 3 hard boiled eggs in a mixing bowl and mash them into small, even pieces with a fork.
Add the mayonnaise, salt, pepper, and a couple of optional pinches of sugar to the eggs and mix well. Taste and adjust for seasonings.
Cut the reserved egg in half lengthwise and place each half in the center of 2 pieces of bread, cut side-down.
Spoon half of the egg salad around the egg on each piece of bread.
Butter the other 2 slices of bread and place them butter side-down over the egg salad.