Preheat the oven to 350°F. Cut each apple into 8 wedges. Separate the crescent roll dough into triangles along the perforated edges. (You should have 16 triangles and 16 apple wedges.)
Place one apple wedge on the widest part of a dough triangle. Tuck the edges in to cover the apple. Roll up, like you would a crescent roll and cover the apple wedge completely. Repeat with all the apples and dough.
Stir together the sugar, cinnamon, and nutmeg in a bowl.
Dip the dumplings in the butter and then in the sugar mixture, turning to coat completely. Place the coated dumplings in single layer in a lightly greased cast-iron skillet or 3-quart baking dish.
Stir together any remaining butter and sugar mixture. Stir in the vanilla. Drizzle the mixture over the dumplings in the pan. Pour the soda in between the dumplings.
Bake, uncovered, for 30 to 35 minutes or until the dough is golden brown and cooked through and the sauce is bubbly. Serve warm with vanilla ice cream.