Combine cookies and butter in a food processor. Pulse a few times until cookies are crumbled and cookie crumbs and butter are combined.
Spoon mixture into individual serving glasses. Set aside.
In a medium saucepan over medium heat, combine sugars, cornstarch, and salt. Whisk to combine. Whisk in milk and egg yolks. Continue stirring until mixture just barely comes to a boil and thickens, about six to eight minutes. (It might take more or less time—watch your mixture carefully!)
As soon as the pudding starts to bubble and thicken to the consistency of pudding, remove from heat.
Stir in white chocolate and butter until melted and smooth.
Pour pudding into glasses on top of crumb base. Serve warm or chill in refrigerator for two hours.
Can be made up to three days in advance; cover with plastic wrap.
Garnish with grated milk chocolate.