|Melt butter in saucepan and bring to a simmer. Beat together egg yolk, lemon juice, and water.
Pour hot butter into egg mixture, beating constantly until thickened. If it doesn't thicken, pour it into the saucepan used to melt butter and heat gently, stirring constantly. Don't heat too much or egg will cook, making the sauce grainy.
Meanwhile, add remaining egg white to whole egg. Fry in butter to your preference. Place egg on plate, pour sauce over it.
This makes just enough sauce for one.