Bring a saucepan of water to a boil and gently lower in your eggs. Let them cook for about 8 minutes so the yolks stay soft and creamy rather than fully firm. Once done, transfer them to a bowl of cold water for about 10 minutes before peeling.
Place your peeled eggs in a mixing bowl and break them up first with a fork. Once they're roughly broken down, switch to a potato masher and mash until smooth. The finer you mash, the creamier your final filling will be.
Add the mayonnaise, mustard, chives, pepper, and salt to the mashed eggs. Stir gently until everything is evenly combined. Taste it before adding any extra salt, since eggs don't need much at all.
Lay out your bread slices and spread a thin, even layer of soft butter on each one. This small step keeps the bread from soaking up moisture from the filling and adds a lovely richness to every bite.
Spoon the egg filling generously onto four of the bread slices, spreading it all the way to the edges. This ensures every bite has that creamy filling instead of just plain bread around the edges.
Top each sandwich with the remaining bread slices. Trim off the crusts if you like a neater look, then slice into rectangles, triangles, or fingers depending on how you plan to serve them.