Make the sauce: combine the shallot and butter in a saucepan over medium heat and cook until butter is bubbling and the shallot is translucent. Add the floor and stir to heat the flour and coat it all with butter. (If using unsalted butter, and a pinch of salt here.)
Turn the heat to high and add the milk, whisking continuously until the milk comes to a simmer and thickens. Reduce the heat to medium-low and cook, skimming any skin that may form on the surface, for about 15 minutes and up to 45, then remove from the heat and cover. This can sit like this for up to 4 hours.
Preheat the oven to broil.
Put the pan over medium heat.
Make the sandwiches by spreading each of the first pieces of bread with Dijon on the unbuttered side, then divide the ham then the cheese between the two. Top with the remaining piece of bread, buttered side up.
Put the sandwiches in the pan and cook till the bread is golden brown. Flip the sandwiches and cook till the bread is golden brown and the cheese is melted.
While the sandwiches are cooking, bring the sauce back up to a simmer.
Heat a nonstick pan with vegetable oil over medium heat and crack two eggs into it. Cook them gently till the white is set for a sunny-side up egg.
While the eggs are cooking, ladle 2 to 3 ounces of sauce over each sandwich, and put the pan below the broiler to gratin the sauce (brown it). When it has lightly browned, remove the pan and put an egg on each sandwich (if you prefer your eggs more cooked, return the pan to the broiler and cook them as you wish).