In a medium saucepan, bring water to a boil. Carefully add the eggs and boil for 10 minutes. Remove the eggs and place them in a bowl of ice water to cool for about 5 minutes.
Once cooled, peel the eggs and chop them into small pieces.
In a mixing bowl, combine the chopped eggs, mayonnaise, Dijon mustard, salt, and pepper. Mix until well combined and creamy. If desired, stir in chopped chives or green onions for added flavor.
Spread the softened butter on one side of each slice of bread.
Place two slices of bread, buttered side down, on a clean surface. Divide the egg mixture evenly between the two slices. Top with the remaining slices of bread, buttered side up.
Heat a skillet over medium heat. Once hot, add the sandwiches and cook for about 3-4 minutes on each side, or until golden brown and crispy.
Remove from the skillet and let cool for a minute before slicing in half. Serve immediately.