For the cake: Preheat the oven to 350 degrees F. Butter three 8-inch round cake pans and line the bottoms with parchment. Butter the parchment and dust the pans with flour, tapping out the excess.
Whisk together the flour, baking powder and salt in a large bowl. Beat the butter and granulated sugar in a large bowl with an electric mixer on medium-high speed until light and fluffy, about 3 minutes. Reduce the mixer speed to medium and beat in the eggs, one at a time, scraping down the bowl as needed. Beat in the vanilla.
Reduce the speed to medium and add the flour mixture in 3 batches, alternating with the cream, beginning and ending with the flour; beat until just smooth, about 3 minutes.
Divide the batter between 2 of the prepared pans. Bake until the cakes are light golden on top and the centers spring back when pressed, 25 to 30 minutes. Transfer to racks and let cool for 10 minutes. (Leave the oven on for the cookie dough.) Invert the cakes out onto the racks (remove the parchment) to cool completely, about 1 hour. Trim the tops of the cakes with a long serrated knife to make level, if needed.
For the cookie dough: Meanwhile, microwave the flour in a small microwave-safe bowl for 1 minute (see Cook's Note). Remove and stir well. Microwave for an additional minute or until the temperature registers 180 degrees F on an instant-read thermometer. Sift into a medium bowl, then whisk in the baking soda and salt.
Beat the brown sugar, granulated sugar, butter, yogurt and vanilla in a large bowl with an electric mixer on medium-high speed until well combined, about 2 minutes. Add the flour mixture and mix on low speed until well incorporated, about 1 minute. (Turn off the beaters and use your hands to help combine if needed). Fold in the chocolate chips with a rubber spatula or mix them in with your hands. (You should have about 4 cups of cookie dough.)
Press a third of the cookie dough (about 1 1/3 cups) evenly into the remaining prepared cake pan and refrigerate until ready to use. Roll a third of the dough into 12 scant 1-tablespoon balls and place them 1 inch apart on a baking sheet. Bake until the cookies are set and starting to brown around the edges, 10 to 12 minutes. Cool the cookies on the baking sheet for 5 minutes, then transfer to a rack to cool completely, about 25 minutes. Cut the cookies into quarters and halves and reserve for decorating.
Roll the remaining third of the dough into different size balls for decorating; refrigerate until ready to use.
For the frosting: Beat the butter in a large bowl with an electric mixer on medium speed until smooth and fluffy, 1 to 2 minutes. Gradually beat in the confectioners' sugar until smooth. Add the vanilla, increase the speed to medium-high and beat until thick, about 2 minutes. Add the milk and beat until the frosting is spreadable, about 1 minute.
Assemble the cake: Place one cake layer on a cake stand or platter and spread with 1/2 cup frosting. Invert the refrigerated cookie dough layer onto a flat surface, gently tapping the bottom of the pan to help release it. Remove the parchment and place the cookie dough on top of the cake layer. Spread 1/2 cup frosting over the top and add the second cake layer. Reserve 1/2 cup frosting, then cover the top and sides of the cake with the remaining frosting. Use an offset spatula to make vertical lines in the frosting around the sides of the cake. Spoon the reserved frosting in a mound in the center of the top of the cake.
Use the cookie quarters to create a border around the bottom of the cake, pressing them into the cake. Fill the spaces between the cookie pieces with small balls of cookie dough. Press two rings of semisweet chocolate chips around the cake above the cookie border. Press two rings of mini chocolate chips above the regular chips.
Decorate the top of the cake with the remaining cookies and cookie dough balls, nestling them on top of and around the mound of frosting. Serve immediately, or refrigerate up to overnight.
When measuring flour, we spoon into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.) Microwaving the flour to 180 degrees F may reduce the risk of foodborne illness associated with eating it raw or undercooked.
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