For the cake: Spray an 8-inch square cake pan with cooking spray, then line with parchment, allowing the parchment to hang over the sides. Spray the parchment paper with cooking spray to prevent sticking. Preheat the oven to 350 degrees F.
Place the cookies and baking powder in a food processor and process until fine crumbs form. With the food processor running, slowly drizzle in the milk. Continue to process until you have a smooth batter, about 2 minutes.
Pour the batter into the prepared cake pan and bake until a toothpick inserted in the center comes out clean, 20 to 22 minutes.
For the icing: Meanwhile, combine the powdered sugar and milk in a large bowl. Whisk until smooth, then set aside.
Allow the cake to cool slightly in the pan before carefully removing it from the pan using the parchment overhang and placing it on a platter.
Allow to cool completely, about 30 minutes. Pour the icing over the top and spread it all over. Top with the sprinkles, then slice and serve.