How to Make It
Make the béchamel. In a medium saucepan, heat the butter and onions over medium-low heat until onions are translucent.
Add the flour and stir until smooth.
Over medium heat, cook until the mixture turns a light, golden sandy color, about 6 to 7 minutes.
Add the milk to the butter mixture 1 cup at a time, whisking continuously until very smooth.
Bring to a boil. Cook 10 minutes, stirring constantly, then remove from heat and run through a sieve.
Season with salt and nutmeg.
Make the French toast. Preheat oven to 375°F.
Mix up the French toast dredge.
Lay out your slices of bread. Layer the cheese on the bread, then béchamel, then prosciutto.
Dip sandwich in dredge and place on a hot flat top or griddle. Cook for 4 to 5 minutes. Flip carefully to keep sandwich together and cook another 3 to 4 minutes. Once both sides are golden brown, place sandwich on sheet pan and place in oven.
Once the sandwich is in the oven, crack a farm egg on the griddle, and cook sunny side up.
Remove the sandwich from the oven once melty and gooey, about 5 to 6 minutes.
Place the sandwich on a plate, and place the egg on top.