Arrange the oven rack to the lower third of the oven cavity. Place baking sheet on rack. Preheat oven to 425ºF (218ºC).
Roll pie crust out, on a very lightly floured surface, into a 12-inch circle. Fit pie crust in an 8-inch standard pie plate. Fold edges under, and decoratively crimp or pinch the edges. Chill until crust is firm, at least 1 hour.
Meanwhile, in a stainless steel skillet, not a non-stick or dark-colored bottom skillet, melt butter (1/2 cup), stirring often, until the butter browns and becomes a light amber color and has a nutty fragrance. Do not let the color of the butter go beyond light amber brown in color, it's very easy for it to burn so don't take your eyes off of it while cooking. Remove the skillet from heat and pour into a small bowl, using a rubber spatula to get every golden drop!!
Line the pastry with aluminum foil or parchment paper. Fill to the top with dried beans or pie weights. Bake at 425°F (218ºC), on the arranged oven shelf for 4 to 5 minutes. Remove weights and foil; bake for 2 more minutes or until golden. Cool on a wire baking rack. Reduce oven temperature to 350ºF (177ºC).
In a mixing bowl, add sugar (2 cups), cornmeal (2 tablespoons), flour (1 tablespoon), and salt (1/4 teaspoon); Whisk until combined.
Add beaten eggs (4) and whisk until well combined.
To the sugar/flour mixture, add cooled browned butter, milk (1/4 cup), vinegar (1 tablespoon), and vanilla (1 teaspoon) to the sugar mixture and blend. Pour into cooled par-baked pie shell.
Place pie into a wire rack and cool preheated oven and bake until edges and center are set, about 45-55 minutes.
Transfer the pie to a wire rack and cool for a minimum of 30 minutes before slicing and serving.