Whisk the eggs and the egg yolk in a bowl until frothy. Add the milk and vanilla; beat well. Slowly stir in the flour, salt, and brown sugar. Pour the mixture into the pie crust.
Whisk together the light whipping cream and the heavy cream in a separate bowl; chill in refrigerator.
Bake the pie in the preheated oven for 10 minutes. Lower the temperature setting to 350 degrees F (175 degrees C). Bake until the center is set, another 35 to 45 minutes. Allow to cool on a rack. Serve at room temperature on a plate in a pool of the chilled cream mixture.