Mix sugar, corn starch and salt thoroughly in heavy saucepan OR top of a double boiler. Whisk milk, egg yolks, and corn syrup together; add to sugar mixture and stir well to combine.
Heat over medium heat, stirring constantly until mixture thickens, about 8 to 10 minutes. Remove from heat and stir in vanilla extract and butter.
Slice bananas into bottom of pie crust. Pour filling over bananas; let cool 10 minutes on a wire rack. Cover with plastic wrap and chill several hours or overnight before serving. Top with whipped cream, if desired.