Heat 1 tbsp oil in a non stick pan.Break an egg into the pan and fry sunny side up.
Sprinkle a little salt and gently transfer it onto a plate. Make 3 more fried eggs similarly and keep them on separate plates.
Mix white sauce and grated cheese in a bowl.Arrange all the bread slices on the table top and apply mustard sauce over 4 of them.
Cut ham slices to the size of the bread slice and keep them over 4 bread slices.
Spread the cheese-white sauce mixture over the ham on 2 bread slices.Cover each with one of the remaining plain slices.
Heat a sandwich griller, drizzle a little olive oil over and place the sandwiches over it, drizzle a little more olive oil over them and close the griller lid and grill till golden on both sides.
Place a cheese slice over the ham on each of 2 of the remaining bread slices.
Cover with the remaining 2 bread slices and grill similarly.
Place a fried egg on each sandwich and serve immediately.