For the dumplings: Trim and discard the stems from the mushrooms and finely chop the caps. In a large saute pan, heat 2 tablespoons of the vegetable oil over medium-high heat. Add the mushrooms, working in batches to avoid overcrowding the pan. Cook until golden brown, 5 to 8 minutes. Add salt to taste. Transfer to a large bowl to cool.
To the same pan, add 1 tablespoon vegetable oil and heat over medium-high heat. Add the kale and cook, stirring, until wilted and most of the liquid has evaporated, 6 to 8 minutes. Add salt to taste. Transfer to the bowl with the mushrooms.
Finely crumble the tofu into the bowl with the mushrooms and kale. Add the cilantro, scallions, soy sauce and sesame oil and mix well. Add salt to taste.
Cover a baking sheet with plastic wrap and dust with cornstarch. Wet the edge of a dumpling wrapper with water. Place about 2 teaspoons of the tofu-mushroom filling in the center and fold to form a half-moon shape. Press the edges to seal well. Place the sealed dumpling on the prepared baking sheet. Repeat with the remaining filling and wrappers.
Place a medium saute pan over medium-high heat and add 1 tablespoon vegetable oil. Working in batches, add the dumplings in a single layer. Add about 1/4 cup water and cover. Cook until the water has evaporated, the bottoms of the dumplings are golden and the wrappers are fully cooked and somewhat translucent, about 5 minutes.
For the dipping sauce: In a small bowl, whisk together the soy sauce, vinegar and sugar until the sugar is dissolved. Add the ginger and sesame oil and stir to combine.
Serve the dumplings with the dipping sauce.