1. In heavy 3-quart saucepan, combine sugar, flour, and salt; stir in milk until well blended. Cook over medium heat, stirring frequently, until mixture has thickened and boils. Remove from heat.
2. In medium bowl, with wire whisk, lightly beat eggs; stir in about 1/2 cup of hot milk mixture. Over medium heat, slowly pour egg mixture back into milk mixture in saucepan, stirring rapidly to prevent curdling. Remove from heat. Strain custard through sieve into large bowl. Press plastic wrap onto surface of custard. Refrigerate until well chilled, about 3 hours or up to overnight.
3. Add heavy cream and vanilla to custard. Freeze in ice-cream maker as manufacturer directs.