Mix brown sugar and flour together in a saucepan. Add cream, maple syrup, 1 egg, and melted butter; stir well to combine. Cook over medium heat, stirring occasionally, until thick, about 20 minutes. Remove batter from heat; cool to room temperature, about 15 minutes.
Line a pie pan with 1 crust; pour in batter. Top with second crust; crimp edges together. Beat remaining egg in a bowl; brush top of pie with beaten egg.
Bake in the preheated oven until golden brown, about 20 minutes.