1. Whisk together brown sugar, flour, and salt in a saucepan. Add one cup of the milk and whisk until smooth. Bring mixture to a boil over medium heat, stirring constantly. Continue to stir until mixture is smooth and thickened, about two minutes. Remove from heat.
2. In a heatproof bowl, beat egg yolks with the remaining one cup milk. Temper the yolks by adding the warm milk mixture to the eggs, a little bit at a time, stirring after each addition. Repeat until about half of the milk mixture has been added and egg mixture is warm to the touch. Pour into saucepan and stir to combine.
3. Bring to a boil over medium heat, stirring constantly; reduce heat to low and simmer until mixture has the consistency of thick pudding, about one minute. Remove from heat and stir in butter and vanilla. Let cool about five minutes or until warm—not hot—to the touch.
4. Pour filling over chilled cookie dough, leveling the top with an offset spatula. Filling should reach about half an inch from the top of the crust.
5. Refrigerate until set, at least three hours.