Preheat the oven to 350°F. Grease three 8 inch round cake pans and line the bottoms with parchment paper.
Make the cookie dough by beating 1/2 cup butter with 1/2 cup granulated sugar and 2/3 cup brown sugar until light and fluffy.
Add 1/4 cup milk and 2 tsp vanilla bean paste to the cookie dough mixture and mix until smooth.
Add 2 cups flour and 1/2 tsp salt to the cookie dough mixture and mix on low just until combined, then fold in 1 cup mini chocolate chips. Chill the cookie dough while you make the cake.
In a medium bowl whisk together 3 cups flour, baking powder, and 1 tsp salt for the cake layers.
In a large bowl beat 1 1/2 cups butter and 2 cups granulated sugar until very light and fluffy, about 3 to 4 minutes.
Add the 3 eggs to the cake batter one at a time, beating well and scraping the bowl after each addition, then mix in 2 tsp vanilla bean paste.
Add the dry ingredients to the cake batter in three additions, alternating with the whole milk and Greek yogurt, beginning and ending with the dry ingredients, mixing just until combined.
Fold 1 cup mini chocolate chips into the cake batter, then divide the batter evenly among the three prepared pans.
Pinch or scoop small pieces of the chilled cookie dough and gently press them into the surface of the cake batter in each pan, spacing them evenly.
Bake the cake layers for 28 to 32 minutes, until the tops are lightly golden and a toothpick inserted in the center comes out with a few moist crumbs.
Cool the cakes in the pans for 10 minutes, then turn them out onto wire racks and let them cool completely.
For the cookie dough buttercream, beat 1 1/2 cups butter until smooth, then gradually add the powdered sugar and mix until fluffy.
Add 1 cup flour and 3/4 cup brown sugar to the buttercream and beat until well combined and no dry streaks remain.
Add 1/4 cup milk, 2 tsp vanilla bean paste, and 1/2 tsp salt to the buttercream and beat until creamy, then fold in 1/2 cup mini chocolate chips.
Place one cooled cake layer on a serving plate and spread a generous layer of cookie dough buttercream over the top.
Crumble or place some of the remaining chilled cookie dough over the buttercream layer, then top with the second cake layer and repeat the filling.
Place the third cake layer on top and spread a thin crumb coat of buttercream over the entire cake, then chill for 20 minutes.
Apply a final thick coat of cookie dough buttercream around the chilled cake, smoothing the sides and top as desired, then chill again while you make the ganache.
For the ganache, place the chopped chocolate in a heatproof bowl. Heat the heavy cream until steaming but not boiling, then pour it over the chocolate and let sit for 2 minutes.
Stir the chocolate and cream together until smooth and glossy, then mix in the vegetable oil until fully combined and slightly fluid.
Let the ganache cool for a few minutes until it thickens slightly, then pour it over the top of the chilled cake, nudging it gently toward the edges to create drips down the sides.
Chill the finished cake for at least 30 minutes to set the ganache, then let it sit at room temperature for 30 to 45 minutes before slicing and serving.