Place ham in the bowl of a food processor with the mustard, Tabasco, paprika, parsley sprigs, grated onion and a few grinds of pepper. Process until very finely chopped and combined.
Melt butter in a small saucepan over medium heat. Whisk in flour and cook 1 minute. Whisk in milk and season with salt, pepper and nutmeg. Thicken to coat spoon.
Arrange bread on a baking sheet and spread the deviled ham evenly over the slices. Top each with 4 asparagus spears then cover with béchamel sauce and some cheese. Transfer to oven and bake until golden.
Cook eggs to your liking and top each sandwich with an egg. Serve with a few cornichons and extra mustard at the table.