Peel the hard boiled eggs and cut them in half. I went to the extra step of separating the yolks from the whites, but will skip it next time. If you want, separate the yolks into a small bowl and mash them up. Chop the whites (or the whites and yolks.)

Melt the butter in a saucepan, then mix in the flour with a whisk, cooking a few minutes until it just lightly browns. Watch it carefully. Slowly whisk in the milk and cook a few minutes, stirring, until thickened. Season with salt, pepper, and cayenne. Mix in the egg (whites or whites and yolks).

Make your toast and butter it. Spoon the sauce over the toast (and sprinkle the yolks over the top if you didn't mix them in with the whites.)