Preheat oven to 350°F and place the unbaked pie crust in a 9-inch pie pan, crimping the edges as desired.
In a large bowl, whisk together the sugar, cornmeal, and flour until well combined.
In a separate bowl, lightly beat the eggs, then add the melted butter, lemon juice, lemon zest, milk, and vanilla extract, stirring gently to combine.
Pour the wet ingredients into the dry ingredients and stir just until combined, being careful not to over-mix.
Pour the filling into the prepared pie crust and place on a baking sheet to catch any drips.
Bake for 40-45 minutes until the center is set but still slightly jiggly, and the top is golden brown.
Remove from oven and let cool completely on a wire rack before slicing and serving.