Preheat the oven to 400 degrees F. Line a baking sheet with foil or parchment paper.
Place the asparagus on the sheet and drizzle with the olive oil. Sprinkle with salt and pepper. Roast for 20 minutes, until the spears are tender.
While the asparagus is roasting, make the cheese sauce. Heat a small saucepan over medium heat and add butter. Once sizzling, whisk in the flour and cook until golden and fragrant, about 1 to 2 minutes, to create a roux. Add the milk and stir constantly until slightly thickened, then reduce the heat to low and stir in the fontina. Stir until melted, then whisk in the salt, pepper and nutmeg.
The asparagus should be finishing now, so you can use the same sheet pan. Place the slices of bread on the sheet. You can spread just 2 or all 4 of the slices with Dijon mustard if you wish – it’s your call. Layer 3 slices of prosciutto each on 2 slices of bread. Top the prosciutto with a few asparagus spears. On the other 2 slices, drizzle on some of the fontina cheese sauce. Sprinkle that with a bit of the shredded gruyere. Sandwich the pieces together to create 2 sandwiches.
Spread the tops of both sandwiches with more Dijon mustard. Drizzle on more of the fontina sauce.
Place the sandwiches in the oven for 10 to 15 minutes, until the cheese is all melty and bubbly. While the sandwiches are baking, fry your eggs and season them with a pinch of salt and pepper.
To finish, you can broil the sandwiches for the last minute or so to create that golden bubbly cheese, but keep your eye on it!
Top the sandwiches with any remaining asparagus spears. Top that with the fried egg. Devour!