Pulse the flour, confectioners' sugar, and salt together in a food processor. Add the butter and pulse until the butter ranges in size from that of a small pebble to that of cornmeal. Empty the flour mixture into a bowl, and crack the egg on top. Lightly scramble the egg and then, using your hands, work it into the dough until the dough starts to come together. Roll the dough into a ball, and then flatten it into a disc about 4 inches around. Wrap it in plastic wrap and refrigerate it for at least 30 minutes.
When you are ready to cook the tart, preheat the oven to 375 degrees. Heat the butter in an 10-inch oven-proof skillet over medium heat. When the butter has melted, sprinkle the sugar evenly over the pan. Add the pears cut side down with their fatter sides towards the outside of the pan. Squeeze in as many as you can because they will reduce slightly in size as they cook. Trim one pear half to fit in the middle.
Turn the heat up to high and cook the pears until the sugar turns a rich, golden brown, about 10 minutes. Rotate the pan if needed to ensure that the sugar browns evenly. If the sugar begins to get darker than a medium brown, turn the heat down slightly. Remove the pan from the heat, and, using two forks, carefully turn the pears over so that their cut sides face up. Return them to the high heat and cook until the sugar is a deep caramel brown, about 5 minutes.
While the pears are cooking, roll the dough out into a circle that is about an inch larger than the diameter of your pan. When the pears are done, remove the pan from the heat and carefully drape the dough over the pan. Taking care not to burn your fingers, tuck the edge of the dough along the side of the pan. If you have measured accurately, the dough should just fit with a bit extra to slip down into the sides. If you do what I usually do and roll it a bit too big, just squeeze it into the pan. Don't worry if the dough is a bit lumpy because this will be the underside of the tart when you are done. Place the pan in the oven and let it cook until the crust is lightly browned, 25 to 30 minutes.
Remove the pan from the oven and let the tart cool for 15 minutes or more. Place a large serving plate over the top of the pan. With oven mitts or pot holders, pick up the pan with the plate on top and quickly flip the pan over so that the tart inverts onto the plate. I recommend that you do this over a surface that is easy to clean (or the sink) because more than once a bit of caramel has oozed out of the side as I flip the pan. Serve warm or at room temperature.
If you make the tart in advance and would like to serve it warm, you can reheat it in a warm oven -- say 250 to 300 degrees -- just before serving.