In a medium saucepan, bring water to a boil. Carefully add the eggs and boil for 10 minutes. Once done, transfer the eggs to an ice bath to cool for about 5 minutes.
Peel the cooled eggs and place them in a mixing bowl. Mash the eggs with a fork until they are crumbly.
Add mayonnaise, Dijon mustard, salt, and pepper to the mashed eggs. Mix until well combined. If using, stir in chopped green onions and paprika for extra flavor.
Spread softened butter on one side of each slice of bread.
Divide the egg mixture evenly among two slices of bread, spreading it to the edges. Top with the remaining slices of bread, buttered side up.
Heat a skillet over medium heat. Place the sandwiches in the skillet and cook for 2-3 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.