To make the filling: In a large mixing bowl, combine the chopped shrimp, oyster sauce, sesame oil, and cornstarch until the shrimp becomes slightly sticky. Add the grated ginger and scallions, then mix until evenly combined.
To assemble the blanket dumplings: With the heat off, spread the cooking oil evenly over the bottom of a large nonstick pan. Scoop 9 portions of the shrimp filling into the pan, leaving a little space between each one. Lay a wonton or dumpling wrapper over each mound like a little blanket, gently pressing it so it hugs the filling.
To pan-fry the dumplings: Place the pan over medium heat and pan-fry for about 1 minute, until the shrimp filling begins to set.
To steam the dumplings: Carefully pour the water into the pan. While you add the water, partially cover the pan to help contain the hot oil splatter. Once all the water has been added, cover the pan completely and steam for about 4 minutes, or until the shrimp is fully cooked through.
To crisp the dumplings: Remove the lid and continue cooking for about 1 minute, until the water has evaporated and the bottoms are beautifully golden and crispy.
To finish and serve: Turn off the heat and sprinkle the scallions over the dumplings. Serve with the soy sauce and chili oil, or drizzle them directly over the dumplings.