Set up a fryer, or heat 2-inches oil in a heavy-bottomed saucepan, fitted with a high-heat thermometer, to 360 degrees F.
To make the filling: Use cheesecloth to squeeze water out of tofu by tightly wringing and squeezing with your hands, or place tofu in a colander lined with 2 damp paper towels and press down with a heavy bowl or pot on top to squeeze moisture out. Pour tofu into a mixing bowl, add remaining filling ingredients and use hands to mix well. Fill a small bowl with water. Place 1 to 2 teaspoons of filling in the center of a dumpling wrapper. Wet the outer edge and fold to make a half-circle. Pinch dough together to seal, pushing air out as you work. Repeat process until the remaining dumplings are all assembled. In batches, deep-fry dumplings until golden, about 3 minutes, stirring occasionally. Remove from the oil and drain on paper towels. Transfer dumplings to a serving platter or bowl and serve with the dipping sauce.
Whisk all the ingredients in a serving bowl to combine and serve with the dumplings.