For the cookie cake:

1. Preheat your oven to 350°F (175°C). Grease and line a 9-inch round cake pan with parchment paper, or use a non-stick baking spray.
2. In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
3. In a large mixing bowl, beat the softened butter with the granulated sugar and brown sugar until creamy and smooth, about 2-3 minutes.
4. Add the eggs, one at a time, beating well after each addition. Stir in the vanilla extract.
5. Gradually add the dry ingredients to the wet mixture, mixing until just combined.
6. Gently fold in the rainbow sprinkles and white chocolate chips (if using), ensuring they are evenly distributed in the dough.
7. Spread the cookie dough evenly into the prepared cake pan, smoothing the top with a spatula.
8. Bake for 18-20 minutes, or until the edges are lightly golden and a toothpick inserted into the center comes out clean.
9. Let the cookie cake cool in the pan for about 10 minutes, then transfer it to a wire rack to cool completely.

For the frosting:

1. In a medium bowl, beat the softened butter with an electric mixer until creamy and smooth.
2. Gradually add the powdered sugar, 1/2 cup at a time, and beat until combined.
3. Add the milk, vanilla extract, and a pinch of salt. Continue to beat until the frosting is light, fluffy, and spreadable. If the frosting is too thick, add more milk a tablespoon at a time until desired consistency is reached.
4. Once the cookie cake has cooled, spread the frosting evenly over the top of the cake using a spatula.

To serve:

1. Garnish with additional rainbow sprinkles for a colorful, festive touch.
2. Slice and enjoy!