Ice cream cake is the ultimate birthday fare whether you're five or fifty- five. But what if that coveted classic had layers of chocolate chip cookie dough hidden inside? You just might have the mother of all birthday cakes. That's not to say this cake can't celebrate other occasions, like Mother's Day, Memorial Day, or even just Monday.
Pour the remaining 1 cup heavy cream into a medium heatproof bowl nested inside a larger bowl of ice water; set aside.
Return vanilla-cream mixture to heat. Stir over medium heat until mixture begins to steam.
In a small bowl, briefly whisk egg yolks. Slowly whisk in some of warm cream mixture, cup at a time, until about half of the cream mixture has been incorporated and egg mixture is warm to the touch. (Do this gradually so that you temper the eggs rather than cook them.)
Pour egg mixture back into saucepan and return to heat. Continue to cook over medium heat, stirring constantly, until mixture thickens slightly and coats the back of a spatula, about 5 to 7 minutes, or until it reaches approximately 165 degrees F to 170 degrees F. Do not let it boil.
Pour mixture through a fine-mesh sieve into the nested bowl of cold cream, straining out solids as well as the vanilla bean. Add vanilla extract and stir until mixture is cool. Cover bowl with plastic wrap, carefully pressing wrap onto the surface to prevent a skin from forming. Refrigerate until completely cool, at least 3 hours or, if possible, overnight.
Preheat oven to 350 degrees F. Butter and line two 9-inch round cake pans with parchment paper. Butter parchment paper.
Carefully run a knife around the inside edge of each pan and invert cakes onto cooling rack. Once they've cooled to room temperature, wrap cakes tightly in 2 layers of plastic wrap and freeze at least 3 hours.
When the ice-cream base has thoroughly chilled, churn in an ice-cream maker according to manufacturer's instructions. Divide in half and transfer to two freezer-safe containers. Freeze 1 to 2 hours, or until ice cream has stiffened slightly but is still soft enough to spread. If ice cream is too firm, let soften briefly until it's spreadable.
Remove one cake layer from freezer and trim to an even 1/2 thick all around. Place it in the bottom of a 9-inch round springform pan and return pan to freezer while you prepare the cookie dough.
For Cookie Dough:
Combine butter and sugars in a large mixing bowl and beat with an electric mixer on medium speed until light and fluffy, 2 to 3 minutes. Add salt, vanilla, and milk; mix well. Slowly add flour and mix until incorporated. Stir in chocolate chips.
Working quickly, remove springform pan from the freezer and gently spread half of the dough over the first cake layer, followed by half of the ice cream. Return pan to freezer while you level the second cake layer. Place second layer on top of ice cream in pan, then spread on remaining cookie dough and top with final layer of ice cream. Return pan to freezer to freeze completely, at least 3 to 4 hours or, if possible, overnight. If you plan to store the cake for more than 24 hours, cover pan with aluminum foil to prevent