Prep the pan and oven: Heat the oven to 350°F (175°C). Grease a 9x13-inch baking pan and line the bottom with parchment for easy removal.
Set aside.
Mix dry ingredients: In a bowl, whisk flour, baking powder, and salt. Stir in the finely crushed cookies. This keeps the crumbs from sinking.
Whip the egg whites: In a clean bowl, beat egg whites on medium-high until soft peaks.
Gradually add half the sugar (about 3/8 cup), beating to glossy medium-stiff peaks. This gives the cake lift.
Beat yolks and sugar: In another bowl, beat egg yolks with the remaining sugar until pale and thick, about 2–3 minutes. Mix in vanilla and 2 tablespoons milk.
Combine batter: Gently fold the yolk mixture into the whites with a spatula.
Sprinkle the flour-cookie mixture over the top in two additions, folding just until no dry spots remain. Do not overmix.
Bake: Spread the batter evenly in the pan. Bake 22–27 minutes, until the top is golden and a toothpick comes out clean.
The cake should spring back when lightly pressed.
Cool and poke: Let the cake cool 10 minutes. Poke holes all over with a skewer or fork, especially around the edges and center. Let it cool to just warm.
Make the soak: Whisk sweetened condensed milk, evaporated milk, whole milk, vanilla, and a pinch of salt until smooth.
Soak the cake: Slowly pour the milk mixture over the warm cake in several passes, letting it absorb between pours.
Cover and chill at least 4 hours, ideally overnight, until fully saturated but not mushy.
Whip the topping: Beat cream cheese (if using) until smooth. Add cold heavy cream, powdered sugar, and vanilla. Whip to medium-stiff peaks.
The cream should hold lines but still look glossy.
Finish with cookies: Spread the whipped cream over the chilled cake. Sprinkle chopped cookies and a few fine crumbs on top. For extra flair, stand a few halves upright along the edge.
Serve: Slice cold with a sharp knife, wiping between cuts.
The flavor deepens by day two, and the cookie pieces soften slightly for a cookies-and-cream icebox feel.