Place three eggs in a saucepan and cover by one inch with room temperature water. Bring the water to a boil over medium-high heat and as soon as it reaches a rolling boil, cover the pan and remove it from the heat. Let the eggs cook in the hot water for fifteen minutes. Rinse with cold water and peel immediately.
Meanwhile, bring another saucepan of water to a boil. You'll need just enough water to cover your eggs by about an inch. Gently lower the eggs into the water. Boil for eight minutes. Immediately drop eggs into an ice bath and peel.
Spread a thin layer of butter on all four slices of bread.
Remove egg yolks from three hard boiled eggs. Add to mixing bowl along with scallions, mayo, cream, sugar, and salt. Finely chop egg whites and combine with mixture. Set aside.
Once the soft boiled eggs have cooled, slice them in half lengthwise and set aside.
Place two jammy egg halves open side down, end-to-end along the centerline of a slice of bread. Repeat with two remaining halves on a second slide of bread. Cover the two jammy egg-topped slices of bread with the egg salad mixture. Try to fill in all the gaps and bring the mixture all the way to the edges of the bread. Top with remaining slices of bread.
Using a sharp, serrated knife, remove the two crusts that run parallel to the line of jammy eggs. Be gentle as the bread is somewhat delicate. Eat the crusts. Then, slice the sandwiches into thirds, making your cuts perpindicular to the crustless edges of the sandwich. Arrange the sandwich on a plate, cut sides up. This will expose the jammy yolks and impress everyone at your lunch table.
Serve and enjoy! I like to eat mine with salt and vinegar chips.