This fabulous dessert goes together very quickly, especially if you use store-bought puff pastry or pie crusts. The dessert is at its best when served warm out of the oven with vanilla ice cream, but it will hold for several hours at room temperature.
Roll the puff pastry into a 10-inch circle on a lightly floured surface. Butter a 9-inch tart pan with a removable bottom and carefully fit the pastry into the pan. Trim the dough so that it extends above the rim by 1/2 inch. Fold the excess under so that you create a doubled thickness of pastry extending about 1/4 inch above the rim of the pan. Refrigerate the pastry for 30 minutes.
If you're using sweet pastry dough, roll it out to a 10-inch circle and carefully fit the pastry into the tart pan, following the instructions above.
Place the mascarpone, egg, and almond extract in a small bowl and whisk to blend.
Place the sugar and almonds in a small food processor or blender and process until the nuts are finely ground. Add the sugar-almond mixture to the mascarpone and stir to blend.
Position a rack in the middle of the oven, and preheat to 400 degrees F.
Remove the pastry from the refrigerator and place the tart pan on a rimmed baking sheet. Prick the dough along the bottom and sides with a fork.
Spread the mascarpone mixture over the base of the tart. Arrange the peach slices decoratively in concentric circles or overlapping slices.
Bake until the crust and filling are golden brown, 30 to 40 minutes. Transfer to a wire rack and let cool for at least 15 minutes before serving. Serve warm with scoops of vanilla ice cream, if you like.