Preheat the oven to 350°F and line one 9-inch round cake pan with parchment paper, leaving a two-inch overhang on two sides so you can lift the baked cookie out later.

In a medium bowl, whisk together the flour, baking soda, and salt until evenly combined.

In a large bowl, beat the softened butter, granulated sugar, and brown sugar together for about 3 minutes, until the mixture turns pale and fluffy. Beat in the eggs one at a time, then the vanilla extract, until smooth.

Add the flour mixture to the butter mixture and mix on low speed just until no dry streaks remain. Fold in the chocolate chips by hand with a wooden spoon.

Scrape the dough into the prepared cake pan and press it into an even layer with damp hands, smoothing the top so it reaches every edge of the pan.

Bake for 28 to 32 minutes, until the edges are deep golden brown and the center looks set but still slightly soft; a toothpick inserted two inches from the edge should come out with a few moist crumbs, not wet batter.

Let the cookie cake cool completely in the pan on a wire rack, about 1 hour. It needs to be fully cool and firm before you cut it, or the shape will crumble.

Lift the cooled cookie out of the pan using the parchment overhang and set it on a large wood cutting board. Using a long sharp knife, make two straight cuts through the cookie, one inch apart, running through the center so you remove a single one-inch strip.

Push the two remaining pieces together so the freshly cut edges touch, pressing gently until the round cookie closes into a pointed oval football shape. Pinch the two tips a little tighter if they look blunt.

In a clean bowl, beat the softened butter for the frosting with the cocoa powder until smooth, then beat in the powdered sugar, milk, vanilla extract, and a pinch of salt until the frosting is glossy and spreadable; add an extra teaspoon of milk if it feels stiff.

Spread the chocolate frosting over the entire football shape with an offset spatula, covering the top and sides completely and smoothing it into a rounded dome down the length of the cookie.

Spoon the white frosting, or melted white chocolate, into a small piping bag or a zip-top bag with one corner snipped off. Pipe one long line down the center of the football lengthwise, then pipe five or six short perpendicular lines across it to look like laces.

Chill the finished football cookie cake for 15 minutes so the frosting sets enough to hold its shape, then slice it into wedges with a sharp knife and serve.