In a small bowl, combine the mushrooms and 1/2 cup warm water. Let soak until soft, 5 to 10 minutes. Drain through a strainer set over a small bowl, pressing firmly on the mushrooms to extract the liquid.
Transfer the mushrooms to a food processor, reserving the soaking liquid. Pulse the mushrooms, occasionally scraping down the sides of the bowl, until finely chopped. Transfer to a medium bowl. Repeat with the carrot and edamame. Add the spinach and toss the vegetables to combine.
In a small bowl, whisk the soy sauce, sesame oil, sugar, 1/4 tsp. pepper, 1/4 tsp. salt and 1/4 cup of the mushroom soaking liquid. In another small bowl, stir the cornstarch with 2 tbsp. of the mushroom soaking liquid.
In a wok or large skillet, heat the vegetable oil over medium. Add the ginger and garlic and stir-fry until fragrant, 15 to 30 seconds. Add the vegetables and the soy sauce mixture. Cook, stirring constantly, until heated through, 1 to 2 minutes. Add the cornstarch mixture; stir until thickened slightly, about 30 seconds. Transfer to a bowl; stir in the scallions. Let cool completely.
Line a rimmed baking sheet with parchment and dust with flour. Place 1 wrapper on a work surface; brush the edges lightly with water. Place the wrapper in the palm of your hand, cupping it slightly in the center. Scoop about 2 tsp. of the filling into the center of the wrapper, leaving a 1/2-inch border. Fold the wrapper in half, enclosing the filling. Press the edges firmly to seal.
Pleat the dumpling's sealed edge by folding every 1/4 inch, forming 4 to 5 pleats; press firmly to seal. Transfer the dumpling to the prepared baking sheet; cover with a kitchen towel. Repeat with the remaining wrappers and filling, spacing the dumplings 1/2 inch apart on the baking sheet. 3. In a large nonstick skillet with a lid, heat 2 tbsp. oil over medium-high. Working in batches, add the dumplings, one at a time and seam-side up, to the skillet. Fry until golden, 1 to 2 minutes.
Holding the lid between you and the skillet in order to protect yourself from splatters, add 1/3 cup water, then cover the skillet immediately. Reduce the heat to medium; cook 3 minutes. Adjust the lid so it's slightly ajar (this allows steam to escape). Cook until most of the water evaporates and the dumplings begin to sizzle, 1 to 2 minutes.
Uncover and cook until the dumplings are browned and crispy on the bottom, 2 to 4 minutes. Remove from the heat. When the dumplings stop sizzling, transfer them to a plate. Serve with the dipping sauce.