Combine the sugar and water together in 2-quart saucepan, and bring to boil until the syrup is clear. Set aside.
Prepare dumplings as follows: Beat egg until well beaten. Add flour until you have a mixture that is looser than biscuit dough, but not as loose as pancakes.
Return the syrup to boil, and add the blueberries. When the blueberries come to a boil, add butter, and let the butter melt.
Immediately drop your dumplings on top of syrup with a tablespoon.(Don't put to many at once in saucepan.) Cover and boil gently for 10 minutes. Serve warm with scoop of vanilla ice cream or Cool Whip.