Whisk together the flour, sugar, and salt in a large bowl. Add the butter and mix with a pastry blender or fork until the mixture resembles coarse meal and the butter is the size of small peas.

Add 1 tablespoon of water and continue mixing. Add more water as needed, 1 tablespoon at a time, until a dough forms.

Wrap the dough tightly in plastic wrap. Refrigerate for at least 2 hours.

Remove the dough from the refrigerator. If necessary, let it sit at room temperature for 10 to 15 minutes until slightly softened but still cold.

Lightly flour a work surface. Roll out the dough into a circle about 1/8-inch thick.

Transfer the dough to a 9-inch pie plate. Trim and crimp the edges as desired. Place the pie plate in the refrigerator while you make the filling.

Place the butter in a medium saucepan or skillet. Cook over medium heat until the butter melts and begins to foam.

Continue cooking, stirring occasionally, until brown specks begin to form at the bottom of the pan and the butter has a nutty aroma. Remove the pan from the heat and allow to cool to room temperature. If desired, strain the butter to remove some or all of the browned bits.

Place the eggs in a large mixing bowl, and beat lightly. Add the cooled butter, sugar, brown sugar, milk, and vanilla, and stir until combined. Stir in the cornmeal and salt.

Transfer the filling to the pie crust. Place the pie plate on a rimmed baking sheet to catch any spills. Bake 55 to 60 minutes, or until the crust is browned and the filling is set.