Preheat the oven to 350 degrees . Using a food processor, pulse the flour, butter and salt until coarse crumbs form. Sprinkle 1/2 cup water on the dough and pulse to blend.
Divide the dough into 4 equal pieces and flatten into disks; wrap each one in plastic wrap and refrigerate until chilled, about 30 minutes. On a floured surface, roll each disk out into a 9-inch round.
In a small bowl, combine the brown sugar and cinnamon. Place half an apple, cut side up, in the center of each dough round. Fill each apple with one-quarter of the cinnamon-sugar mixture. Top each with another apple half, cut side down. Pull the dough up around the apple. Pinch the dough together to seal.
Place the dumplings, seam side down, in a lightly greased baking dish and bake until light-golden, about 1 hour. Transfer to plates and spoon any syrup from the dish over the apples. Serve warm with the ice cream.