Preheat the oven to 350ºF. Place pastry on a parchment-lined rimmed baking sheet. With a paring knife, gently score (being careful not to go all the way through) the pastry about one inch from the edge on all sides. Prick bottom of pastry all over with a fork, line center area only with parchment paper and fill with pie weights or beans. Bake for 20 minutes or until the edges are golden. Remove pan from oven and place on a cooling rack. Remove weights and parchment paper.
Meanwhile, heat the oil over medium heat and add the onion. Season with salt and pepper and let sauté until slightly caramelized about 6 to 8 minutes. Remove from heat to cool.
Fill a medium saucepan with water and bring to a boil. Cut the squash crosswise into ¼ – inch thick rounds. Add to the pot of boiling water, cook for 30 – 60 seconds, drain and let dry on a paper-towel lined cookie tray.
In a small bowl, whisk together the ricotta, egg and parsley. Season with salt and pepper to taste and spread onto puff pastry. Top with the onions. Arrange squash pieces in overlapping rows until tart is filled. Bake for 15 minutes. Remove from oven, brush with butter and return to the oven for five minutes longer. Remove pan from oven, sprinkle with feta, and let cool for 10 minutes before serving.