To make the dough: In a medium-sized bowl, whisk together the flour, baking powder, salt and sugar (or whisk the sugar into the self-rising flour)., Cut in the butter until the mixture is crumbly, using your fingers, a pastry fork or blender, or a mixer, leaving some pea-sized pieces., Sprinkle in the smaller amount of milk, and stir. Add a bit more milk as needed, just until the dough comes together., Divide the dough in half, wrap each half in plastic, and put it in the freezer to chill., To prepare the apples: Wash, peel and core the apples, saving the peels., Slice the apples in half around the equator. Sprinkle the 8 apple halves with 1 tablespoon lemon juice (to keep them from browning), then set them aside., To make the syrup: Combine the brown sugar, water, and apple peels in a medium-sized saucepan, bring to a boil over medium heat, and simmer for 10 minutes., Combine the sugar, cinnamon and nutmeg in a small bowl., Preheat the oven to 375°F. Lightly grease a 9" x 13" pan., To assemble the dumplings: Remove half the dough from the freezer and turn it out onto a well-floured surface. Roll it into a 12" x 12" square; it'll be thin., Cut the dough into four 6" squares., Place an apple half on each square, and then sprinkle each with a tablespoon of the cinnamon-sugar mixture. Moisten the edges of the square and bring them up over the apples, tucking them together in the center of the apple., Place the apples, seam-side up, in the prepared pan., Repeat with the remaining dough and apples., Strain the syrup to remove the apple peels. Stir in the lemon juice and boiled cider., Pour this syrup evenly over the dumplings. Sprinkle the dumplings with the sparkling white sugar or demerara sugar., To bake the dumplings: Bake the dumplings for 45 to 50 minutes, until the apples are tender and the pastry is brown. The liquid will bubble and thicken., Remove the dumplings from the oven, and let them sit for at least 10 minutes before serving; this allows the dough to firm up. Serve as is, or with vanilla ice cream, if desired.