In a large pot melt the butter. (I like to melt it on low)
Stir in flour, salt and pepper.
Have your milk ready to pour in slowly. The butter will want to clump up.
Cook on med/high
Stir vigorously or use a whisk. I take a fork and smash large clumps against the spoon or the sides of the pan.
Once all the lumps are out and it is smooth comes the hardest part- stir continuously- great workout for the arms! As it thickens you will feel the spoon start to drag more through the mixture. It won't get super thick, but will be considerably thicker.
Add chopped eggs and cook on low for a few more minutes until all of it is warmed through.
Break 2 pieces of bread onto a plate
Cover with the egg sauce