Bring 1 cup of the milk to a boil in a medium saucepan over medium heat.
Meanwhile, combine the sugar, cornstarch and salt in a medium bowl. Whisk in the remaining 1 cup milk a little at a time to avoid lumps, then whisk in the egg.
When the milk is boiling, remove the saucepan from the heat and gradually whisk in the sugar-milk mixture, stirring constantly. Return to medium-low heat and cook, whisking constantly, until the pudding comes to a simmer and is thick and creamy, 5 to 7 minutes. Remove from the heat and whisk in the butter and vanilla until smooth. Stir in the cookie butter.
Divide the pudding among four to six 1/2-cup ramekins or serving glasses, layering each with 1 to 2 tablespoons of the cookies. Top with additional cookies. Cover with plastic wrap and refrigerate until fully set, about 2 hours.
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