You can make these day-of or up to 6 hours ahead. Just warm them up in the oven for a couple of minutes before serving.
In a large skillet, heat the oil over medium-high. Add the shallot; cook, stirring often, until softened, 2 to 3 minutes. Add the mushrooms; cook, stirring often until softened and beginning to brown, 5 to 6 minutes. Stir in the chopped parsley and garlic; stir for 1 minute. Transfer to a bowl; season with salt and pepper. Let cool. Mix in the cheese.
Preheat the oven to 400°. Line 2 baking sheets with parchment. On a work surface, open the puff pastry. Using a 3 1/2-inch biscuit cutter, cut out 18 rounds; divide between the sheets. Mound the filling on the rounds. Hold 1 round in your hand like a taco. Using a wet fingertip, dampen the edge of the pastry. Fold the pastry over the filling, pressing to seal the edges; return to the baking sheet. Repeat with the remaining rounds.
Brush the pies with the egg. Sprinkle with flaky salt and pepper. Top with the torn parsley. Bake, rotating the sheets halfway through baking, until the pastry is puffed and golden brown, 18 to 20 minutes. Let cool slightly before serving. (The pies can be made up to 6 hours ahead. Warm in a 400° oven for 5 minutes.)
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