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1
In a large bowl, combine the flour, salt, sugar, and sage and mix until well-combined. Using a box grater, grate the frozen butter into the flour, then mix gently with your fingers to incorporate it into the flour, just until the butter forms pea-sized clumps. Add the ice-cold water and creme fraiche, then stir with a wooden spoon until the mixture comes together into a dough. Pat into a disk, wrap in plastic wrap, and place in the refrigerator to chill for at least 1 hour.
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2
Preheat oven to 425u00b0 F. In a medium bowl, whisk together the brown sugar, cinnamon, salt, nutmeg, ginger, and cloves. In a small saucepan, melt the butter until it crackles, turns a deep brown, and smells nutty and fragrant. Set aside.
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3
Using a sturdy, sharp knife, carefully slice the acorn squash in half. Scoop out the insides of the acorn squash and slice into thin, 1/4 to 1/8 inch slices. (I found that a mandoline slicer was very useful for this, though I would use it very carefully.) Toss the slices in the spice mixture, then add the brown butter and toss again until well-coated.
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4
Lay the squash slices in a single layer on a parchment-lined baking sheet and roast in the oven for about 30 minutes, or until squash is tender and caramelized.
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5
While the squash is roasting, make the hazelnut frangipane. Combine the hazelnut meal and the flour in a small bowl, and whisk until well-combined. In a medium bowl, beat the sugar and butter with an electric mixer on medium-high until pale and fluffy, about 2 minutes. Add the egg and beat again. Finally, add the extracts and the hazelnut and flour mixture, then beat on low speed until just combined.
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6
Once the pie dough has had enough time to chill, remove it and separate it into four balls. One ball at a time (keeping the other three chilled), roll the dough out to about a 6-inch circle. I find that rolling the dough is easiest between two lightly-floured pieces of parchment paper, but do whatever you are used to.
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7
In the center of each dough circle, spread 1/4 of the frangipane into about a 5-inch circle. Arrange the roasted and cooled squash slices in an overlapping single layer (or however you like, really) over the frangipane. Fold in the outer edges of the dough over the filling, forming a 1-inch crust all the way around. Return the assembled galettes to the fridge while you work with the remaining ones.
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8
When all four have been assembled, beat an egg with a splash of milk, then brush the crust with the egg wash, sprinkle it with sugar, and bake for 30 to 40 minutes, or until crust is a deep, golden brown.
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9
When ready to serve, use an electric mixer and beat about 1/2 cup creme fraiche on low speed, just until it forms soft peaks (a few minutes was all it took for me). You can add about 1/4 teaspoon of vanilla and 1 to 2 teaspoons powdered sugar just before beating, if you like. Let galettes cool briefly, then serve with whipped creme fraiche on the side.
