Preheat the oven to 350°F. Lightly coat an 8×8-inch or 9×9-inch baking dish with nonstick spray.
Slice the prepared apple into eight even wedges after peeling and coring.
Unroll the crescent dough and place a single apple wedge at the broad end of each triangle. Roll each piece up snugly around the apple.
Arrange the wrapped apple rolls into the greased baking dish with a bit of space between each.
In a small mixing bowl, stir together the melted butter, white sugar, cinnamon, and all-purpose flour until the mixture is just blended.
Distribute this spiced butter mixture evenly over the tops of the crescent-wrapped apples using a spoon.
Pour the lemon-lime soda carefully around the dumplings in the open spaces of the pan, avoiding pouring it directly on top.
Transfer the baking dish to the oven and bake for 35 to 40 minutes, or until the crescent tops are a deep golden brown and crisped.
Serve these dumplings warm, accompanied by whipped cream or a scoop of vanilla ice cream if desired.