Oil an 11x17-inch baking dish with vegetable oil. Line dish with foil, and lightly coat foil with more oil.
In the bowl of a standing electric mixer, soften the gelatin with 3/4°C of the water.
Place the sugar, corn syrup, the remaining 3/4°C water, and the salt in a heavy saucepan. Bring to a boil and cook over high heat until the syrup reaches 234 to 240 degrees F, or the soft-ball stage, on a candy thermometer.
With the whisk attachment of the mixer at full speed, beat the hot syrup slowly into the gelatin until mixture is very stiff, about 15 minutes. Beat in the vanilla. Pour the mixture into foil lined dish. Allow mixture to rest, uncovered, at room temperature 10-12 hours.
Using a fine sieve, sprinkle the confectioners' sugar onto a cutting board. Turn the stiffened marshmallow mixture out onto the sugar, and using a small, lightly oiled cookie cutter, cut into shapes. Be sure to dip the cut edges of the marshmallows into confectioners' sugar to prevent sticking.
Once made, keep in an airtight container - best eaten within two days of making.