Preheat grill to a high heat. Begin by making a béchamel sauce. First melt the butter in a saucepan over a medium heat until it begins to foam and turns a slight brown. Add the flour and stir until mixed, then continue to stir and cook the flour for approximately three minutes.
Gradually add the milk, adding just a little at a time and making sure it is fully incorporated to ensure no lumps form. Once all the milk is added simmer for a further five minutes until a fairly thick texture is reached. When the wooden spoon is taken out you should be able to run a finger across the middle of the spoon and the sauce should not run back into this line, it should remain clear.
At this stage finish the sauce of by seasoning to taste and adding the mustard and just the egg yolk from one of the two eggs.
Lightly toast the bread and butter as you would for toast. Layer the ham and cheese onto the bread before topping with the béchamel sauce. Place the slices of bread back under the grill under the béchamel begins to bubble and brown, and the cheese has melted.
Fry the remaining egg in a separate pan and add to the slices of toast once they are grilled. Top with seasoning and the chopped chives.