Make the Cookie Dough:
In the bowl of an electric mixer, beat the butter, brown sugar, and granulated sugar on medium-high speed until very well combined. Add the egg and vanilla extract, and beat until fully incorporated.
Gradually add the flour and salt on low speed, mixing until just combined. Stir in the chocolate chips.
Shape and Chill the Dough:
Roll the dough into 1-inch balls and place them on a parchment-lined baking sheet. Refrigerate until firm, at least 1 hour (or up to 2 days if stored in an airtight container).
Prepare the Frying Batter:
In a medium bowl, whisk together the flour, baking powder, salt, and sugar. Add the milk and vegetable oil, whisking until the batter is smooth and lump-free.
Fry the Cookie Dough:
Heat 2 inches of oil in a medium heavy pot until it reaches 360°F (180°C).
Take the chilled cookie dough balls, dip each in the batter, ensuring they are fully coated, and allow any excess batter to drip off. Gently drop the dough balls into the hot oil, frying 4-5 at a time.
Cook Until Golden:
Fry each dough ball for 1½ to 2 minutes per side, or until puffed and golden brown. Remove with a skimmer and drain on a paper towel-lined baking sheet.
Repeat the process with the remaining cookie dough balls and batter, keeping the oil between 360°F and 370°F.
Serve Warm:
Drizzle the fried cookie dough balls with melted chocolate and dust with powdered sugar before serving. These are best served warm on the day they’re fried but can be kept in a 175°F oven to stay warm until ready to serve.