Preheat the oven to 350 F and grease a 9 x 13-inch baking dish. Whisk the flour, baking powder, and salt together in a bowl.
Beat the plant butter and granulated sugar until pale and fluffy. Beat in the eggs one at a time, followed by the vanilla.
Add the flour mixture in three additions, alternating with the oat milk. Fold in the 10 crushed chocolate sandwich cookies and spread the batter evenly in the dish.
Bake for 28 to 33 minutes, until the center springs back when lightly pressed. Cool for 15 minutes, then use the handle of a wooden spoon to make evenly spaced holes across the warm cake.
Slowly pour the sweetened condensed coconut milk over the cake, concentrating it in the holes. Cool completely, then refrigerate for 1 hour.
Whip the dairy-free whipping cream, powdered sugar, and vanilla to firm peaks. Crush 10 of the remaining cookies and fold half of the crumbs into the whipped cream.
Spread the cookies and cream topping over the chilled cake. Finish with the remaining crumbs and 4 coarsely broken cookies, then refrigerate for at least 2 hours before serving.