Preheat the oven to 350°F.
Roll out the pie dough, then ease into a 9-inch pie pan.
Whisk together the eggs, then add in the sugar, butter, salt, and vanilla, stirring until well combined and smooth. Pour the batter into the pie pan.
Bake for 45-55 minutes or until golden brown. Check to see that the pie is baked through (it will be gooey but it should be set). If you're making a 6-inch pie (see Recipe Notes), reduce the baking time to 30-40 minutes. If you make one deep dish (instead of two 9-inch pies), then increase baking time slightly.
Allow pie to cool on a cooling rack, then refrigerate to set up further.
An hour before serving, take the pie from the refrigerator and allow to come to room temperature (not strictly necessary, but that's how I like it). Serve with fresh whipped cream or vanilla ice cream, if desired.