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1. FOR THE DOUGH: Process flours, sugar, and salt in food processor until combined, 2 to 3 pulses. Add butter and pulse until it forms pea-size pieces, about 10 pulses. Transfer mixture to medium bowl.
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2. Sprinkle water and vinegar over mixture. With rubber spatula, use folding motion to mix until loose, shaggy mass forms with some dry flour remaining (do not overwork). Transfer mixture to center of large sheet of plastic wrap, press gently into rough 4-inch square, and wrap tightly. Refrigerate for at least 45 minutes.
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3. Transfer dough to lightly floured work surface. Roll into 11 by 8-inch rectangle with short side of rectangle parallel to edge of work surface. Using bench scraper, bring bottom third of dough up, then fold upper third over it, folding like business letter into 8 by 4-inch rectangle. Turn dough 90 degrees counterclockwise. Roll out dough again into 11 by 8-inch rectangle and fold into thirds again. Turn dough 90 degrees counterclockwise and repeat rolling and folding into thirds. After last fold, fold dough in half to create 4-inch square. Press top of dough gently to seal. Wrap in plastic wrap and refrigerate for at least 45 minutes or up to 2 days.
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4. FOR THE FILLING: Cover mushrooms in bowl and microwave until just tender, 3 to 5 minutes. Transfer to colander to drain and return to bowl. Meanwhile, heat 1 tablespoon oil in 12-inch skillet over medium heat until shimmering. Add leeks and thyme, cover, and cook, stirring occasionally, until leeks are tender and beginning to brown, 5 to 7 minutes. Transfer to bowl with mushrooms. Stir in creme fraiche and mustard. Season with salt and pepper to taste. Set aside.
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5. Adjust oven rack to lower middle position, place pizza stone on oven rack, and heat oven to 400 degrees. Remove dough from refrigerator and let stand at room temperature for 15 to 20 minutes. Roll out on generously floured (up to 1/4 cup) work surface to 14-inch circle about 1/8 inch thick. (Trim edges as needed to fo
