Combine gelatin and 1/2 cup cold water in the bowl of an electric mixer with whisk attachment. Let stand 30 minutes.
Combine granulated sugar, corn syrup, salt, and 1/2 cup water in a small heavy saucepan; place over low heat, and stir until sugar has dissolved. Wash down sides of pan with a wet pastry brush to dissolve sugar crystals.
Clip on a candy thermometer; raise heat to high. Cook syrup without stirring until it reaches 244 degrees (firm-ball stage). Immediately remove pan from heat.
With mixer on low speed, slowly and carefully pour syrup into the softened gelatin. Increase speed to high; beat until mixture is very thick and white and has almost tripled in volume, about 15 minutes. Add vanilla; beat to incorporate.
Generously dust an 8-by-12-inch glass baking pan with confectioners' sugar. Pour marshmallow mixture into pan. Dust top with confectioners' sugar; wet your hands, and pat it to smooth. Dust with confectioners' sugar; let stand overnight, uncovered, to dry out. Turn out onto a board; cut marshmallows with a dry hot knife into 1 1/2-inch squares, and dust with more confectioners' sugar.