Before starting, let the butter and cream cheese soften and bring the eggs toward room temperature. Preheat the oven to 350°F (175°C). Grease two 9 inch round pans, line the bottoms with parchment, and flour the sides. Whisk the flour, baking powder, and salt together. Chill the coarse cookie pieces for 10 minutes while mixing the batter. Beat the softened butter and granulated sugar on medium high for 3 minutes, until pale and fluffy. The mixture should look light, with a visibly aerated texture; some sugar grains may remain.

Beat in the eggs one at a time, mixing 30 seconds after each. Stir in the vanilla and scrape down the bowl until the batter looks smooth.

Add one third of the flour mixture, then half the milk, mixing on low speed. Repeat until combined, and mix only until no dry streaks remain.

Gently fold in the chilled cookie pieces with a spatula. The batter should look speckled and thick, and it should fall slowly from the spatula.

Divide the batter evenly between the pans and smooth the tops. Bake at 350°F (175°C) for 25 to 30 minutes, until golden and a toothpick comes out with a few moist crumbs.

Rest the layers in their pans for 10 minutes, then turn them onto a wire rack. Cool for at least 1 hour until they feel room temperature all the way through.

Beat the butter and cream cheese on medium for 2 minutes. Add the powdered sugar, vanilla, and cream, then beat on high for 2 minutes until fluffy. Fold in 1/4 cup of the finely crushed cookies and reserve the remaining 1/4 cup for decoration.

Place one layer on a plate and spread frosting over the top as filling. Add the second layer, then coat the top and sides of the assembled cake with a thin crumb coat. Chill for 15 minutes, apply the remaining frosting, and press the reserved 1/4 cup of cookie crumbs onto the sides.