In a small saucepan, melt 2 Tbsp of butter over medium-low heat. Add the flour and whisk while cooking, until the butter and flour are well combined and foaming, about 2 minutes. Slowly add  the milk, continuing to whisk until the mixture is light in color and smooth and starting to thicken up. Turn off the heat, whisk in the mustard, salt, and freshly cracked pepper. Set aside.

Preheat your oven to 350 F. Grease a 9x13 inch baking dish with the remaining 1 Tbsp of butter.

Slice the croissants in half through the middle (lengthwise) so you have a top and bottom. Lay the bottoms into the pan, squishing them together. Brush each half with a generous layer of béchamel sauce. Any extra, put on the bottom layer.

Next, layer the torn pieces of prosciutto and then the shredded cheese, reserving about 1/4 cup of cheese to add at the end.

Place the tops back onto the bottoms in the baking dish. If they don't fit perfectly, you can rearrange them. They will be squished together.

Whisk together the eggs, milk, white wine, mustard, and a pinch of salt and pepper. Pour the mixture over the croissants. With your hands, press down on the croissants to help them absorb the egg mixture. Top with the remaining cheese. Let sit for 30 minutes (or up to 12 hrs) before baking.