In a medium saucepan, stir together the brown sugar, heavy cream and butter. Cook over medium heat until the sugar dissolves and the mixture comes to a full boil, becoming thick and brown, about 5 minutes. Remove from the heat. Reserve 1 tablespoon of the milk. Slowly add the remaining milk to the sugar mixture.
Whisk together the cornstarch, egg yolks, vanilla and reserved 1 tablespoon of milk in a medium bowl. Add the cornstarch mixture to the sugar mixture over medium-low heat and cook, stirring constantly, until thick, about 3 minutes. Pour the filling into the prebaked pie crust.
Make a meringue by beating the egg whites and a pinch of salt with an electric mixer until they begin to set stiff. Add the granulated sugar and continue beating until the whites are stiff and hold peaks. Spread the meringue on top of the pie, taking care to spread it to the edges of the crust. Bake until light brown, about 10 minutes.