Boil, peel, and chop eggs. Place eggs in a saucepan with water covering an inch above. Bring to a boil, then cover and turn off the heat. Let sit for 12 minutes. Peel under cold running water and chop.
Mix chopped eggs, mustard, mayo, and salt in a bowl. This mixture can be made in advance.
Cut bread into circles using a cookie cutter, avoiding the crust.
Assemble sandwiches: bread, egg salad, another bread circle, and garnish with parsley.