Using a slotted spoon, gently transfer the eggs (8 large) to the water, being careful to avoid cracking. Adjust the heat so the water is boiling but isn't disrupting the eggs.

Set a timer and cook for seven minutes.

Using the slotted spoon, transfer four of the eggs to the ice water. Keep the remaining eggs in the water and boil for five minutes more.

Remove the jammy eggs from the ice water and set them aside. Using the slotted spoon, transfer the hard-boiled eggs to the ice water, adding more ice as needed.

Let the hard-boiled eggs chill in the ice water for at least five minutes.

Rap the chilled first batch of (jammy) eggs all over on the work counter to crack the shells. Starting at the wide end, and working under a stream of cold running water, peel the eggs. Set these eggs aside.

Rap and peel the chilled hard-boiled eggs (the second batch) under a stream of cold running water.

Cut in half and transfer the egg yolks only to a medium bowl and mash well with a fork. Finely chop the egg whites and add to the mashed yolks.

Mix in the mayonnaise (3 Tbsp), scallion (1 Tbsp), mustard (2 tsp), and vinegar (1 tsp). Season with salt ( to taste) and pepper ( to taste).

To make each sandwich, spread the toast with half the mayonnaise.

Spread one slice with half of the egg salad in a mound. Nestle two soft-boiled eggs in the center of the egg salad, with the egg tips running vertically.

Cap with the second toast slice, mayonnaise side down, and press gently to even out the salad.

Place the sandwich on a dinner plate. Using a serrated knife, cut through the center of the sandwich horizontally, which will slice through the whole eggs to reveal the centers.