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1
Rinse shiitake and common mushrooms. Trim and discard shiitake stems; coarsely chop both kinds of mushrooms (keep separate). In a 10-inch frying pan over medium-high heat, melt 1 tablespoon butter with 1 tablespoon olive oil. Add shiitakes and stir often until browned, 4 to 5 minutes. Pour into a bowl. Melt another tablespoon butter with another tablespoon olive oil in pan. Add common mushrooms and stir often over medium-high heat until liquid evaporates and mushrooms are well browned, 10 to 12 minutes. Add to bowl. Cool slightly, then finely chop mushrooms.
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2
In same pan, over medium-high heat, melt third tablespoon butter. Add onion and stir often until beginning to brown, about 8 minutes. Add garlic and stir until fragrant, about 1 minute. Pour in vinegar and stir until liquid evaporates, 1 to 2 minutes. Stir in parsley and the mushrooms; season to taste with salt and pepper.
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3
Combine flour and 1/2 teaspoon salt in the bowl of a food processor. Whirl briefly to mix. Add 6 tablespoons chilled butter and whirl until coarse crumbs form. Pour in 1/4 cup cold water and whirl just until dough comes together. Squeeze a little in your hand; if it doesn't hold together, add another tablespoon or two of water and pulse to mix. Pat dough into a ball, then flatten slightly into a disk, wrap in plastic, and chill until cold, 15 to 20 minutes.
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4
Preheat oven to 375u00b0. On a well-floured surface, with a floured rolling pin, roll dough into an oval 1/4 to 1/8 inch thick. Brush half with beaten egg. Spoon tablespoonfuls of the mushroom mixture into one hand, squeeze tightly, and mound an inch apart on egg-washed side of dough. Fold bare half over mounds, and press between them to seal. With a 2-inch cookie cutter, cut out tarts and transfer to a baking sheet. Gather remaining dough, roll again, fill, and cut out more tarts.
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5
Bake until beginning to brown, about 15 minutes. Brush tarts with egg wash (discard any remaining) and continue baking until deep golden brown, about 5 minutes longer. Serve warm.
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6
Wine pairing: Pick a Pinot Noir with earthy loam and mushroom notes to echo the mushrooms in the tarts.
