Mix sugar and water together in a small bowl until melted. Let cool, at least 15 minutes. Add egg; mix until combined.

Combine 2 cups plus 2 tablespoons flour, baking powder, and baking soda in a large bowl. Mix in sugar-egg mixture and 2 tablespoons oil with a wooden spoon. Knead by hand until dough sticks together, adding 1 teaspoon water at a time if too dry. Wrap dough in plastic wrap; let rest for 30 minutes.

Unwrap dough and place on a lightly floured work surface. Roll out into long tubes. Cut tubes into small pieces and roll pieces into balls about 1/2-inch wide.

Spray balls lightly with water. Place sesame seeds in a shallow dish. Dip dough balls into the sesame seeds and roll to press in the seeds.

Heat deep-frying oil in a large saucepan over high heat. Add dough balls; reduce heat to low. Cook, turning occasionally, until lightly golden brown, 4 to 6 minutes. Drain on paper towels.