Preheat the oven to 350°F (175°C). Lightly butter a 9 inch round cake pan or coat it with baking spray to prevent sticking.
In a medium bowl, whisk together the flour, baking soda, and salt until evenly combined.
In a large mixing bowl, beat the softened butter until smooth. Add the brown sugar and granulated sugar and mix for about 2 minutes until creamy but not fluffy.
Add the egg and vanilla extract. Beat until the mixture becomes smooth and glossy, scraping down the sides of the bowl if needed.
Reduce the mixer speed to low and gradually add the dry ingredients. Mix only until the flour is incorporated into the dough.
Use a silicone spatula to gently fold the chocolate chips into the dough until evenly distributed.
Transfer the dough to the prepared pan and press it evenly into the pan, reaching the edges. Sprinkle a few extra chocolate chips on top.
Bake for 20 to 25 minutes until the edges turn golden while the center still appears slightly soft.
Remove the pan from the oven and allow the cookie cake to cool completely on a wire rack while still in the pan.
For the frosting, beat the softened butter until smooth. Add powdered sugar, cocoa powder, milk, and vanilla extract. Mix on low speed until combined, then beat for 2 minutes until light and creamy.
Carefully remove the cooled cookie cake from the pan and place it on a serving plate.
Transfer the frosting to a piping bag fitted with a decorative tip and pipe a border around the edge of the cookie cake. Finish with colorful sprinkles if desired.