One 8-inch (20cm) round pan, 2 inches deep, buttered
1. For the apple filling, peel, halve, core, and cut the apples into thin wedges. Melt the butter in a wide saucepan or Dutch oven with a cover, and add all the remaining ingredients but the rum. Stir well and place on medium heat. Cook until the apples start to sizzle, then cover the pan and decrease the heat. Cook until the apples have exuded water, between 5 and 10 minutes. Uncover the pan and stir occasionally while the water evaporates. Remove from heat, add the rum, and spread the filling in a shallow bowl. Refrigerate the apple filling.
2. For the dough, combine the flour, sugar, baking powder, and salt in the bowl of a food processor and pulse to mix. (You can also make this by hand, in a bowl.) Cut the butter into 8 or 10 pieces and add. Pulse until the butter is no longer in visible pieces. Add the eggs and pulse until the dough starts to form a ball. Invert to a floured work surface and carefully remove the blade. Wrap and chill the dough.
3. When you are ready to assemble the pie, set a rack in the lowest level of the oven and preheat to 350ºF (180ºC) degrees.
4. Lightly knead the chilled dough to soften it. Roll a little more than half the dough on a floured surface and line the pan. Cut away excess dough at the rim of the pan. Spread the chilled apple filling in the crust. Roll the remaining dough and cut an 8-inch (20cm) disk. Set it atop the filling and fold the dough on the side of the pan over it to seal. Use a fork to press down the folded dough and to pierce some holes throughout.
5. Bake until the dough is baked through and dark golden, about 35 to 40 minutes. Cool on a rack.
6. Unmold the pie to a platter keeping the bottom of the pie as the top. Combine the confectioners’ sugar and water in a small saucepan and stir well. Heat to lukewarm and quickly spread on the pie. Let the icing set before serving.