Add 2/3 cup water and 1/2 cup dark brown sugar to a medium saucepan and cook over high heat until sugar dissolves and mixture comes to a boil, about 3 to 4 minutes.
Reduce heat to medium and add a spoonful of tapioca flour. Cook, stirring constantly until smooth, as like thickening a sauce with cornstarch. Add half the remaining flour and stir until a sticky dough forms. Remove from heat and add remaining flour. Stir until dough comes together to form a ball (not all flour will be incorporated yet).
Turn dough out onto a clean surface and knead until all the flour is incorporated and dough is smooth. Add more flour or water as needed.
Divide dough into smaller portions, keeping dough covered in plastic wrap while working so that it doesn't dry out. Roll each portion into a thin rope, approximately 1/4-inch thick. Cut each rope into 1/4-inch pieces.
Dust hand with a small amount of flour and roll each piece into a ball. Place in a tray with tapioca flour to coat and prevent balls from sticking to each other. Work quickly as it gets harder to roll the balls as they cool.
For the last couple portions, it may be necessary to put them in the microwave briefly to warm the dough back up, no more than 10 seconds.
Put pearls in a fine mesh sieve and shake to remove excess flour.
Note: Alternatively, you can purchase quick cooking, premade tapioca pearls in your local Asian supermarket or online. Cook according to manufacturer's instructions and add to brown sugar syrup as described below. I use WuFuYuan Black Sugar Flavored 5 Minute Quick Cooking Tapioca Pearls.