In a 3 1/2 quart crockpot, combine blueberries, the 1/2 cup granulated sugar, water and tapioca. Cover and slow cook until the berries have formed a thick sauce, 5 to 6 hours on low.
In a medium bowl, whisk until blended, flour, remaining 2 tablespoons granulated sugar, baking powder and salt. Using a pastry blender, two knives or cool fingertips that move lightly and quickly, cut in the butter until the mixture resembles coarse oatmeal. In a small bowl, beat milk and egg together. Stir into flour mixture to form a soft dough.
Turn crockpot setting to high. Drop the dough by tablespoonfuls on top of the blueberries. Cover and cook until the topping is firm and a toothpick inserted in the center comes out clean. This will take about 30 minutes.
Sprinkle the dumplings with the brown sugar. Let the dumplings stand for 5 minutes before serving, then spoon into individual bowls.