Oil the bottom and sides of a 13- by 9- by 2-inch rectangular metal baking pan and dust with confectioners’ sugar, banging out any excess sugar.
In the bowl of a standing electric mixer, sprinkle the gelatin over the cold water and let stand to soften.
In a medium saucepan, combine granulated sugar, corn syrup, hot water and salt and cook over low heat, stirring with a wooden spoon, until sugar is dissolved.
Once dissolved, increase the heat to medium and boil mixture, without stirring, until a candy or digital thermometer registers 240°F (about 12 minutes).
Remove pan from the heat and pour the sugar mixture over the gelatin mixture. Mix on low until the gelatin is dissolved.
Continue mixing on high speed until the mixture is white, thick and has nearly tripled in volume. This should take about 6 minutes if using standing mixer, and about 10 minutes if using a hand-held mixer.
In a large bowl, beat the egg whites until they just hold stiff peaks.
Add the egg whites and vanilla to the marshmallow mixture and beat, by hand, until well combined.
Pour the mixture into the prepared baking pan and sift ¼ cup confectioners’ sugar evenly over the top. Chill, uncovered, until firm, at least 3 hours and up to a day.
After chilling, run a thin knife around the edges of the pan and invert pan onto a large cutting board. Lifting up one corner of the inverted pan, use your fingers to loosen the marshmallow “pillow” until it releases onto the cutting board. Using a knife or a shaped cookie cutter, cut into squares, circles or any desired shape.
Sift additional confectioners’ sugar over the sticky edges.
These will retain their freshness at room temperature for about 24 hours. Keeping them chilled until you are ready to use them will prolong their freshness a bit. But we bet they’ll be long gone before they have a chance to get stale!