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Preheat oven to 350º.
Cut ¼" from bottom of potatoes (so they won't roll around) and remove ends. Slice each potato crosswise to create ¼" fans without cutting completely through potatoes. The best way to do this is to place chop sticks, pencils, wooden spoon handles or anything long enough and about ¼" thick on either side of the potato to keep from cutting all the way through.
Rinse potatoes under running water*, let drain, and transfer to a microwavable plate. Microwave all four at once until slightly soft to the touch, about 8 minutes, flipping potatoes halfway through cooking. Set aside until cool to the touch. In a bowl, mix together the sour cream, milk and nutmeg. Set aside. In another bowl, mix together the Parmigiano-Reggiano Cheese (shredded/shaved), panko bread crumbs and finely chopped Prosciutto. Set aside. Grate the butter with a large hole grater (cold - leave it in the freezer for an hour or so before prep time - and leave it in fridge afterwards until you're ready)**.
Arrange potatoes, sliced-side up, in a 9x11 baking dish, casserole dish or lasagna pan. Cut Gruyere cheese into approximately 2" squares about ¼" thick and insert into the cuts of each potato. Pour milk/sour cream mixture over the top of all the potatoes, making sure to get some in each of the cheese-stuffed slices. Now sprinkle evenly the Parmigiano/Panko/Prosciutto mixture over the tops of each of the potatoes. There's no benefit to wasting any of this in the milk bath. Try to keep all of it as potato topping. And finally, sprinkle the butter over the top of everything.
Bake until crisp and potatoes are beginning to brown, 35 to 40 minutes. Remove potatoes from oven, garnish with chopped scallions and serve.
Enjoy.
* Moisture + starch + heat = Library Paste! If you don't rinse the segments they will be cemented together.
** You can cut the butter into little ½-¼" cubes (a lot of work), or you can use melted butter and mix it in along with Parmigiano/Panko/Prosciutto mixture. But, that would be a lot messier.
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