In a medium bowl, whisk together flour, sugar, baking powder, and salt.
Add the milk, egg, and vanilla. Whisk until smooth. The batter should be thick but pourable — thicker than pancake batter, thinner than dough.
Pour oil into a deep pot or saucepan, filling it about halfway. Heat to 350–360°F (175–180°C).
Working with a few cookies at a time, dip each Oreo fully into the batter. Use a fork to lift it, letting excess batter drip off. Lower it into the hot oil.
Fry for 30–45 seconds per side, flipping once, until golden brown and puffed. Remove with a slotted spoon and place on paper towels.
While still warm, dust generously with powdered sugar. Serve immediately.
