On a lightly floured surface, roll puff pastry to an 1/8-inch-thick 12-inch circle. Using a plate or cake mold as a guide, cut out an 11-inch round. Refrigerate until ready to use.
Peel and core the pears, and slice them into quarters. Set aside.
For the caramel sauce
In a 10-inch oven-safe skillet, melt the white sugar and brown sugar on low-medium heat until light brown. It will be clumpy at first until it starts to dissolve into a syrupy liquid (about 5 to 10 minutes).
Remove from heat and add the butter, stirring for 1 to 2 minutes until incorporated.
Arrange the pears on top of the caramel sauce, while being careful not to touch the hot caramel. Place the skillet in the oven and bake for 10 to 15 minutes, depending on the firmness of the pears: less firm equals less time and more firm equals more time.
Remove the skillet from the oven. Scoop out any of the excess liquid and save for later. Set skillet aside.
Remove the dough from the refrigerator. Drape the chilled pastry over the pears and caramel sauce. Tuck the edges into the pan and prick with a fork.
Crack the egg into a bowl and whisk. Brush the pastry with the egg wash and bake for 20 to 25 minutes or until the crust is golden brown.
Remove from the oven and cool for 5 minutes. Run a knife around the edge of the crust, place the serving plate upside down over the top of the skillet, and carefully flip the tart onto the plate. Brush the pears with the remaining syrupy liquid and serve warm with whipped cream or ice cream.
Variations of this recipe can also be made with apples, quinces, and peaches.