Preheat oven to 350 degrees F (175 degrees C). Line a 9-inch baking pan with parchment paper, leaving an overhang on 2 sides.
Pulse 12 cookies into fine crumbs in a food processor. Add melted butter; mix until crumbs are coated. Press crumbs into the prepared pan. Wipe out food processor bowl.
Bake crust in the preheated oven until set, about 5 minutes. Remove from oven and let cool. Keep oven on.
Place cream cheese, sour cream, sugar, egg, and vanilla extract into the food processor; pulse until creamy. Add 1/2 cup white chocolate chips to the food processor and pulse a few times to evenly distribute. Pour cheesecake filling over the cooled crust. Smooth out the top using a spatula.
Bake in the preheated oven until cheesecake is mostly firm when pan is gently shaken, about 35 minutes. Cool on a wire rack for 10 minutes.
Break up remaining 12 cookies. Press cookies and remaining 1/4 cup white chocolate chips gently into the top of the cheesecake. Let cool to room temperature, about 30 minutes. Cover with plastic wrap and chill until set, at least 3 hours. Lift out of the pan and cut into squares before serving.