1. Peel, core, and cut apples into 1-inch-thick wedges to make about 6 cups.
2. Select a frying pan, with an ovenproof handle, that measures 10 inches across the top and holds 2 1/2 to 2 3/4 quarts. In pan, mix sugar and 3/4 cup water, then add apples and butter. Cook over high heat, shaking pan often, until sugar melts, then boil until apples begin to caramelize, 20 to 25 minutes.
3. On a lightly floured board, roll pastry into a 10-inch-wide round, trimming edges to make even. Lay pastry over apples in pan and brush lightly with beaten egg.
4. Bake apple tart in a 375° oven until crust is deep golden brown and juices bubble around the edge, 15 to 20 minutes.
5. Invert a rimmed serving plate over tart. Holding pan and plate together, turn dessert over. Lift off pan and fit any apples that are stuck to pan back onto dessert. Cut tart into wedges and transfer to dessert plates. Add ice cream.
6. Pour Calvados into frying pan over medium heat. When hot, ignite Calvados (away from flammable surfaces or vent) and spoon flaming brandy over servings.