For the pudding: Put the eggs yolks in a medium bowl. Whisk together the milk, sugar, cornstarch and salt in a medium saucepan, and bring to a low simmer over medium heat, whisking constantly to prevent sticking on the bottom. Once the mixture comes to a simmer, continue whisking until it is just thicker than heavy cream, 3 to 4 minutes. Remove the saucepan from the heat, pour about 1 cup of the mixture into the egg yolks and whisk to combine. Pour this milk-egg mixture back into the milk mixture in the saucepan. Return the saucepan to medium heat, and whisk until very thick, 3 to 4 minutes. (When you drag the whisk through the thickened pudding, the streaks should hold firm for a few seconds.)
Whisk the butter and vanilla into the pudding, and transfer it to a large bowl to cool. Set the bowl in some ice water and whisk frequently to help the pudding cool faster.
For the raspberry topping: While the pudding cools, stir together 2 tablespoons water and the cornstarch in a small bowl. Heat the raspberries and sugar in a medium skillet over medium heat, stirring occasionally, until the raspberries thaw and break down and the sugar dissolves completely, 3 to 4 minutes. Stir in the cornstarch mixture, and bring to a simmer, stirring constantly; continue cooking and stirring until the mixture thickens enough to coat the back of a spoon, 1 to 2 minutes. Remove the skillet from the heat, and let the mixture cool completely.
Build the pie: Spread 1/2 the cooled pudding into an even layer in the pie crust. Scatter the peaches over the pudding, then the cookies over the peaches. Finish with an even layer of the remaining pudding. Cover with a piece of plastic wrap, and refrigerate for 1 hour. Spread the raspberry topping over the pudding. (Stir the topping first to smooth out if needed.) Refrigerate the pie until the pudding is completely set, about 2 hours. Slice and serve.
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