Place flour, salt, and sugar in the bowl of a food processor; pulse 2 or 3 times to mix. Add butter; process until mixture resembles coarse meal, about 10 seconds. With machine running, add 4 to 6 tablespoons ice water in a steady stream through feed tube; process just until mixture begins to hold together, no more than 30 seconds. Flatten dough into a disc, and wrap in plastic. Chill dough at least 1 hour.
Combine all syrup ingredients in a medium saucepan. Cook over medium-high heat until liquid has reduced by half and has slightly thickened, about 20 minutes. Remove from heat; set syrup aside.
Combine Armagnac and cherries in a small bowl; let sit 20 minutes. Drain cherries, reserving Armagnac; add Armagnac to reserved syrup. In another small bowl, combine cherries, flour, ground almonds, brown sugar, butter, nutmeg, mace, and cinnamon; mix filling well with a wooden spoon.
Peel apples; remove stems and top 3/4 of core, leaving 1/4 inch intact at bottom. Rub each apple with a lemon quarter to prevent discoloration. Divide filling among four apples, filling cavities. Insert a cinnamon stick into each cavity. Wrap exposed end of cinnamon stick with a piece of heavy-duty aluminum foil. Set filled apples aside.
Heat oven to 450 degrees. Turn out dough onto a lightly floured work surface. Roll dough to an 1/8-inch thickness, and trim to a 14-inch square. Cut 4 squares, 7 inches each, from dough. Brush entire surface of one square lightly with beaten egg white; place a filled apple in the center. Bring 2 adjacent corners of dough together at the tops, and press together to create a triangular flap. Repeat, making four flaps. Lightly brush one side of each flap with egg white, and press firmly against the apple. Repeat with remaining dough and apples. Transfer to a plate, and chill 15 minutes.
Mix together the egg yolk and heavy cream. Brush dumplings lightly with the egg wash, and transfer dumplings to a roasting pan, spacing dumplings at least 1 inch apart.
Bake 15 minutes. Reduce oven temperature to 375 degrees. Remove roasting pan from oven, and pour the reserved syrup over dumplings. Return pan to the oven, and bake 10 minutes more, basting twice. Continue baking dumplings until deep-golden brown, about 10 minutes more. Transfer pan to a wire rack to cool slightly. Serve dumplings with the warm syrup.