Form wrappers from the dough. Work with half the dough at a time.
Before assembling the dumplings, line a baking sheet with parchment paper. Dust the parchment paper with cornstarch or flour.
For each dumpling, hold a wrapper in a slightly cupped hand. Scoop up about 1 tablespoon of filling with a bamboo dumpling spatula, dinner knife, or fork and position it slightly off-center toward the upper half of the wrapper, pressing and shaping it into a flat mound and keeping about ½ to ¾ inch of wrapper clear on all sides.
Fold, pleat, and press to enclose the filling and create half-moon, pea pod, or pleated crescent shapes. Place the finished dumplings on the prepared baking sheet. Repeat with the other wrappers, assembling the dumplings and spacing them apart so that they do not touch. Keeping the finished dumplings covered with a dry kitchen towel, form and fill the wrappers from the remaining dough.
To pan-fry the dumplings, using a medium or large nonstick skillet; if both sizes are handy, cook two batches at the same time. Heat the skillet over medium-high heat and add 1½ tablespoons of oil for a medium skillet and 2 tablespoons for a large one. Place the dumplings one at a time, sealed edges up, in a winding circle pattern. The dumplings can touch. Fry the dumplings for 1 to 2 minutes, until they are golden or light brown on the bottom.
Holding the lid close to the skillet, use a kettle or measuring cup to add water to a depth of roughly ¼ inch. Cover with a lid or aluminum foil, lower the heat to medium and let the water bubble away for 8 to 10 minutes until it is mostly gone. After 6 to 8 minutes, move the lid or foil so that it is slightly ajar to allow steam to shoot out from underneath. This lessens the drama of condensation dropping down onto the hot oil when you remove the lid. When most of the water is gone, remove the lid. Let the dumplings fry for another 1 to 2 minutes, until the bottoms are brown and crisp.
Remove from heat and serve immediately. Display them with their bottoms facing up so that they remain crisp. Serve with dipping sauce.