Step 1Cut the butter into thick pieces (about 2 tablespoons each) and place in the freezer for 10 minutes to harden slightly. In a small bowl or measuring cup, whisk together the salt and water until the salt is dissolved. Place the mixture in the freezer for 10 minutes to chill.
Step 2Meanwhile, on a large, flat surface, spread the flour into a fairly large rectangle about one-fourth inch thick. Place the cold butter pieces on top, and sprinkle a little of the flour on top of each piece.
Step 3Use a rolling pin to press down on the pieces of butter, flattening them and rolling over them, to create sheets of fat. Use a bench scraper to gather the flour and butter sheets together and repeat to continue flattening the butter. Eventually, the butter and flour will begin to look like dried-out flakes of paint.
Step 4Gather the mixture again into a pile, and form a well in the center. Pour the cold salt water in the center of the well, along with the sugar. Slowly incorporate the flour and butter sheets into the well, working from the outside of the flour pile with the bench scraper and watching that the water does not escape from the well. Continue working just until the mixture comes together to form a dough. The dough will look a little dry at first but, once chilled, will come together.
Step 5Form the dough into a disk and wrap tightly in plastic wrap. Chill the dough for at least 2 hours.
Step 1Heat the oven to 375 degrees. To cook the potatoes, fill a large stockpot halfway with water and bring to a boil, and lightly salt the water.
Step 2Meanwhile, heat a saute pan over medium-high heat until hot. Melt 2 tablespoons butter, then add the mushrooms. Saute the mushrooms in the butter until softened and lightly colored, about 6 minutes. Season with a pinch of salt and remove from heat, placing the mushrooms in a bowl.
Step 3In a large saucepan, melt the remaining three-fourths cup butter over medium-low heat. Add the flour and whisk to create a roux; continue to cook gently for a few minutes until it begins to turn a blond color. While whisking, gradually add the milk to form the bechamel, careful to beat smooth any lumps. Continue whisking until the bechamel sauce comes to a gentle simmer and begins to thicken. Simmer very gently, stirring frequently, for about 10 minutes. Remove from heat, add the goat cheese and sage, and whisk until smooth. Season with a generous one-fourth teaspoon salt and one-fourth teaspoon pepper, or to taste. Keep the sauce in a warm place; if it gets too thick, add some milk to achieve a pourable consistency.
Step 4Slice the potatoes into disks about one-third-inch thick using a mandolin or knife. Add the potatoes to the pot of boiling water and allow it to return to a simmer. Lower the heat and gently simmer until the potatoes are still a bit firm in the center, about 5 minutes; drain in a large colander.
Step 5Place a layer of potatoes in the bottom of a 3 1/2- to 4-quart pan or rondeau, and spoon some bechamel over just to cover. Place the sauteed maitake mushrooms over the layer of bechamel. Add another layer of potatoes, topping that with enough bechamel to cover completely.
Step 6On a lightly floured surface, roll the galette dough to a disk wide enough to cover the gratin in the pan, about one-half-inch thick. Place the galette dough over the bechamel sauce, and brush with the egg wash. Slice a few vent holes in the dough.
Step 7Place the pan in the oven and bake until the filling is bubbly and the top is a nice golden-brown, about 1 hour. Set the gratin on a rack to cool 10 minutes before serving.