Preheat the oven to 375 degrees F.

Line a baking sheet with parchment paper. In a medium skillet, over medium heat, melt the butter. Add the onions. Season with salt and pepper. Saute until the onions start to wilt, about 4 minutes.

Stir in the garlic and continue to cook for 1 minute. Remove from the heat and cool completely. In a mixing bowl, combine the onions and cheese. Mix well. Season with salt and pepper.

With a 2 1/2-inch cookie cutter, cut 24 rounds 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 onion mixture on each round, leaving a 1/8-inch border.

Bake until lightly browned, about 15 minutes. Remove from the oven and sprinkle with the parsley. Serve warm.