Prepare the sauce: Whisk all the ingredients together in a small bowl until the cornstarch is dissolved; set aside.

Prepare the veg: Slice mushrooms, thaw green beans if from frozen or cut into 1 inch pieces if fresh, peel and coin carrot ~1/4 inch thick, grate ginger, and slice green onions.

Start by dry-sauteeing mushrooms and then set them aside.

Heat 2 tablespoons of the oil in a large nonstick frying pan over medium-high heat until shimmering. Place the potstickers or gyoza pleat-side up in a single layer. Fry undisturbed until the bottoms are light golden-brown, about 3 minutes.

Carefully add the water (be careful because the oil may spatter). Cover and cook for 3 minutes. Uncover and, using tongs, turn the dumplings onto their unpleated sides. Continue cooking uncovered until all the water has evaporated, the filling is cooked through, and the unpleated sides are golden-brown, about 3 minutes more. Transfer to a large plate.

Add in more oil, the mushrooms, green beans, carrot, and salt. Stir-fry until crisp-tender ~2 minutes; Add the ginger and stir-fry for 30 more seconds.

Return the reserved potstickers and any juices back to the pan and toss gently to combine. Whisk the sauce again to recombine and dissolve the cornstarch, then add to the pan. Stir-fry until the sauce thickens and coats everything in the pan, about 30 seconds. Transfer to a serving dish and sprinkle with the scallions. Serve immediately.