Preheat oven to 375F. Butter 2 round 9 cake pans. Dust each pan with flour, tapping out excess.
Make basic yellow cake batter, add chopped cookies and mix just until blended. Divide batter evenly between pans and bake for 25-30 minutes or until golden brown, and cake tester inserted in center comes out clean.
Allow cakes to cool in pans on a wire rack for 15 minutes. Remove cakes from pans and continue to cool on wire racks.
Make fluffy white icing:
In a medium saucepan over medium-high heat, whisk together milk and flour until mixture comes to a boil and thickens to a paste. Remove from heat and allow to cool to room temperature.
In a bowl of electric mixer, cream butter, sugar and vanilla until light and fluffy. Add cool milk and flour mixture and beat at medium-high speed until fluffy. Gradually add confectioner's sugar 1 tbs. at a time and beat until fully incorporated.
Place on cake top-side up on a serving platter, and spread about half of icing over layer completely covering sides and top. Sprinkle top of layer with about 1/4 c. ground cookie crumbs. Top with second cake top-side up and cover top and sides with remaining icing. Spread cookies crumbs over entire cake.
You can decorate with entire chocolate sandwich cookies on top of the cake.