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Wipe the mushrooms clean and use a spoon to scrape out the gills on the underside of the caps. Roughly chop the mushrooms into 1" pieces.
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In a medium skillet over high heat, heat the oil until it shimmers. Add the mushrooms, reduce the heat to medium and cook, stirring occasionally, until the mushrooms are soft (4 to 5 minutes). Season with salt and pepper.
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Remove the mushrooms from the heat and let cool to room temperature. When cooled, chop finely and place in a medium bowl.
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Stir the goat cheese and cumin into the mushrooms. Set aside.
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Place a rack in the center of the oven and preheat to 350°F. Line two baking sheets with parchment.
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On a well-floured surface, roll the puff pastry 1/8" thick, making a rectangle about 28" x 12". Using a 3 1/2" to 4" round cutter, cut out 24 circles, re-rolling the scraps as needed.
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Brush the edges of the pastry circles with the egg wash. Place a rounded teaspoon of filling on one half of each circle, leaving the edge of the circle uncovered. Carefully stretch and fold the dough over the filling to make a half-moon. Press the edges together firmly and seal with the tines of a fork.
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Transfer the empanaditas to the prepared baking sheet. Refrigerate for 15 minutes to allow the dough to rest.
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Brush the tops of the empanaditas with the egg wash and bake them for 35 to 40 minutes, until the edges where the pastry has risen are golden brown. Remove them from the oven and let them cool for at least an hour before serving. Pico de gallo or salsa verde are nice partners for these pastries.
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Store leftover pastries, well wrapped, in the refrigerator for up to five days.