Add butter and onions to a medium skillet over medium-low heat. Cook until the onions are soft and translucent, about 6 minutes. Increase heat to medium, add the sweet potato and parsnip. Cook, stirring occasionally, for 5 minutes—then add water, stir, and continue cooking until the potato and parsnips are tender, about 10 minutes more. Stir in the mushrooms, rosemary, and salt and cook until the mushrooms soften, about 5 minutes. Remove from heat.
On a lightly floured surface, roll out the puff pastry sheet to ⅛-inch thickness. Cut the dough into rounds using a round cookie cutter or jar lid (3 inches wide) and rerolling the dough scraps when necessary. You should be left with approximately 18 rounds. Transfer rounds to parchment lined baking sheets.
Place 1 heaping teaspoon of the squash mixture in the center of each round. Fold the dough over so it forms a half moon and, with a fork, press down on the edge to seal. Repeat. Place pies 1-inch apart on prepared baking sheets. Brush tops with the egg mixture and bake for 30 minutes.
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