Peel and finely dice the shallots. Place a large, heavy bottomed saucepan over medium low heat. Add the butter and olive oil to the pan. Add the diced shallots to the plan along with a sprinkle of kosher salt. Sauté the shallots, stirring regularly, for about 15 minutes until they are tender and look translucent.
Add the diced apples along with 3 tbsp of the apple cider vinegar, the brown sugar, and 1 tsp kosher salt, stirring to combine. Continue cooking, stirring regularly, until the shallots are dark brown and deeply caramelized. Keep an eye on the shallots to make sure they don't burn on the bottom of the pan. If the shallots begin to stick, add a splash of water to loosen them. Do this as needed until the shallots are meltingly tender.
When the shallots are caramelized add the remaining tablespoon of apple cider vinegar along with the whole grain mustard. Cook for another few minutes then taste to check the seasoning. Remove from the heat and let cool. The apple shallot jam can be stored in the fridge.
Croque Monsieur
To make the béchamel sauce, melt the butter in a medium saucepan over medium high heat. Add the flour, whisking until smooth to make a roux. Slowly add the milk to the roux in batches, whisking constantly to prevent lumping. Heat the béchamel sauce until it thickens and bubbles, then turn down the heat. Add the Dijon mustard, salt, nutmeg, and pepper and whisk to combine.
Preheat the oven to 400°F. To assemble the sandwiches, arrange half the bread slices on a parchment lined baking sheet. Spread a thick layer of the apple shallot jam on the bread slices. Top with the ham and half of the shredded Gruyère cheese. Spread a heaping tablespoon of the béchamel on each of the remaining slices of bread. Flip the bread over so that the side with the béchamel is inside the sandwich. Press down gently to secure the sandwich. Top each sandwich with a thick layer of béchamel followed by the remaining Gruyère cheese.
Bake the croque monsieur for about 15 minutes until the sandwiches are heated through and the cheese is melted inside. Turn the oven to broil and set on high. Broil the sandwich for about 2 to 3 minutes until golden brown and bubbling. Top with chopped fresh chives if desired and enjoy!