Combine the gelatin and 1/2 cup of cold water in the bowl of an electric mixer fitted with the whisk attachment and allow to sit while you make the syrup.
Combine the sugar, corn syrup, salt, and 1/2 cup water in a small saucepan and cook over medium heat until the sugar dissolves. Swirl pan while heating but do not stir. Raise the heat to high and cook until the syrup reaches 240 degrees on a candy thermometer. Remove from the heat.
With the mixer on low speed, slowly pour the sugar syrup into the dissolved gelatin. Put the mixer on high speed and whip until the mixture is very thick and glossy white, about 15 minutes. Add the vanilla and mix thoroughly.
With a sieve, generously dust an 8 by 12-inch nonmetal baking dish with confectioners' sugar. Pour the marshmallow mixture into the pan, smooth the top, and dust with more confectioners' sugar. Allow to stand uncovered overnight until it dries out.
Turn the marshmallows onto a board and cut them in squares. Dust them with more confectioners' sugar.