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Preheat the oven to 350°F with the rack in the center position. Line a 12" round pan that’s at least 2" deep with parchment and lightly grease.
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To make the cookie cake: In a large bowl, combine the sugars and melted butter. Whisk gently to combine, then vigorously for 30 seconds. The mixture will become thicker and slightly lighten in color.
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Once combined, add the egg, egg yolks, and vanilla and whisk to combine.
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Add the flour, salt, and baking powder to the bowl and stir with a flexible spatula until almost combined. Add the chocolate chips to the bowl and stir until no flour streaks remain.
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Transfer the dough to the prepared pan and spread into an even layer.
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Bake the cookie cake for 22 to 26 minutes, or until it is evenly golden brown across the top and pulls away from the sides of the pan slightly.
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Remove from the oven and cool on a wire rack for 25 minutes before turning out onto a wire rack to cool completely (about 1 hour) while you make the frosting.
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To make the frosting: In the bowl of a stand mixer fitted with the flat beater attachment (or in a large bowl working with an electric hand mixer) beat together the butter and salt on medium-high speed until smooth.
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Reduce the mixer speed to low (or pause if you're working with an electric hand mixer) and add the confectioners’ sugar, 4 teaspoons (19g) of the milk, and the vanilla. Mix on low speed until the confectioners' sugar is incorporated, then increase the speed to medium-high and beat until fluffy. If the frosting seems thick or dry, add milk 1/2 teaspoon at a time until the frosting is soft and fluffy.
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Remove 3/4 cup (210g) of the frosting and place it in a small bowl. To the remaining frosting, add the cocoa powder and 1/2 teaspoon of the milk, then continue beating until evenly incorporated. (Add milk 1/4 teaspoon at a time, if needed, until the frosting is soft and fluffy.)
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Decorate the chocolate chip cookie cake as desired (see "tips," below for details on how to achieve the swirled two-tone effect); garnish with the sprinkles of your choice, if desired.
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Storage information: Store any leftover cookie cake, covered, at room temperature for up to 4 days; freeze for longer storage.
