Preheat the oven to a stable temperature of 350 degrees Fahrenheit.
In a large, sterile mixing bowl, combine the eggs, buttermilk, and vanilla extract. Whisk these ingredients together until they achieve a consistent and homogeneous texture.
In a separate mixing bowl, blend the granulated sugar, cooled melted butter, cornstarch, and salt. Ensure that these ingredients are thoroughly mixed, achieving an even distribution throughout.
Gradually combine the contents of the two bowls, stirring continuously to ensure a smooth and uniform mixture.
Gently transfer this mixture into the unbaked pie crust, taking care to distribute it evenly within the crust.
Place the pie in the preheated oven and bake for approximately 55 minutes. It is important to note that the actual time may vary depending on the specific characteristics of the oven.
Once the pie has been baked to a golden brown perfection, remove it from the oven. Allow the pie to cool to room temperature.
This pie may be served either chilled or at room temperature, based on personal preference.