In a medium bowl, whisk the pudding mix and milk together until combined and thickened. Stir in the crushed Oreos; set aside.

In the chilled bowl of a stand mixer fitted with the whisk attachment, whip the heavy cream, 1/2 cup powdered sugar, and vanilla together until stiff peaks form. Transfer whipped cream to a separate bowl, and return bowl to mixer.

Switch the whisk attachment out for the paddle attachment. Beat the cream cheese and 1/3 cup powdered sugar until combined. Add the Irish cream and continue beating until smooth. Fold in 2 cups of the whipped cream.

Fold 1 cup of the whipped cream into the chocolate Oreo pudding.

To assemble, start by lining the bottom of an 8-inch square dish with half (16) of the Oreos. Spread half of the pudding evenly on top of the cookies. Top with dollops of the Irish cream filling, then spread into an even layer. Repeat the layering with the remaining cookies, pudding, and Irish cream filling. Spread the rest of the whipped cream on top, then sprinkle with crushed Oreos.

Cover and refrigerate overnight to allow the cookies to soften up for easier cutting. Serve with a drizzle of chocolate syrup, if desired.