Heat oven to 350°F and place a rack in the center.
Fit the pie shell into a 9-inch pie plate, crimp the edge, and chill while you mix the filling.
Whisk sugar, cornmeal, flour, and salt in a large bowl until evenly combined.
Add eggs one at a time, whisking well after each until smooth and glossy.
Whisk in lemon zest, lemon juice, vanilla, and milk until the mixture looks silky.
Slowly whisk in the cooled melted butter until fully incorporated. Tap the bowl to release bubbles.
Place the chilled crust on a sheet pan. Pour in the filling. Shield crust edge with foil if desired.
Bake 45–55 minutes until the top is lightly browned and the center has a gentle jiggle.
Cool on a rack until room temperature, then chill at least 2 hours before slicing for neat pieces.