Preheat the oven to 350°F and grease a 9 x 13-inch baking dish to prevent sticking.
Melt the butter in a medium saucepan over low heat. Once melted, whisk in the brown sugar, vanilla, and rum (if using), until the sugar is melted and the sauce is smooth. Set the sauce aside to cool while you prepare the dumplings.
Peel, core and slice the apples into large pieces (about 1/2 to 1-inch wide.) In a large bowl, toss the apple slices with the cinnamon.
Roll out the crescent roll dough and separate into the crescent pieces. Place two apple slices on the wide bottom of each piece and roll up the apples in the dough, pinching the edges to close.
Arrange the dumplings in the prepared baking dish. Pour the sauce over the dumplings, evenly coating. Then, pour the club soda over the dumplings.
Sprinkle the tops of the dumplings with extra brown sugar. Bake for 20 minutes, then increase the temperature to 375°F and bake for an additional 10 to 15 minutes, until the dumplings are golden brown and the apples are tender.
Serve the dumplings warm with a drizzle of sauce from the baking pan and a scoop of vanilla ice cream or whipped cream.