Preheat the oven to 375ºF. Line a baking sheet with parchment paper.
In a medium-size skillet, melt the butter over medium heat. Add the onions, season with 1/8 teaspoon of the salt and 1/8 teaspoon of the black pepper, and cook, stirring occasionally, for 2 minutes. Add the mushrooms and season with the remaining 1/2 teaspoon salt and 1/2 teaspoon black pepper. Cook, stirring, for 2 minutes. Add the garlic and cook, stirring, for another 2 minutes. Remove the skillet from the heat and pour the mushrooms into a medium-size mixing bowl. Add the goat cheese and 1/2 cup of the Parmigiano and stir to blend.
With a 2 1/2-inch cookie cutter, cut 24 rounds out of the puff pastry. Place them on the parchment-lined baking sheet, prick each round randomly with the tines of a fork, and brush with the beaten egg. Spread a tablespoon of the mushroom mixture on each round, leaving a 1/8-inch border. Bake until lightly browned, about 15 minutes.
Remove from the oven, sprinkle the rounds with the remaining 1 tablespoon Parmigiano and the parsley, and let cool for several minutes before serving.
Yield: 8 servings (3 tarts each)