Mash 2 bananas and put in a medium saucepan with 1 cup milk and maple syrup. Heat over medium heat until hot.

Whisk ¼ cup milk with arrowroot or cornstarch. Whisk into the banana mixture. Continue to cook, stirring, until thick. Remove from heat and stir in vanilla and butter. Let cool to room temperature. Can be refrigerated if desired - transfer to a bowl, cover with plastic wrap, placing the wrap directly on the pudding surface, and refrigerate for up to 2 days.

Before serving, coarsely crush three-quarters of the cookies. Slice the remaining banana.

Layer the pudding into 4 dessert bowls. Layer pudding, crushed cookies, and sliced bananas. Top with whipped topping, whole cookies, and banana slices.