Sift the powdered sugar and cornstarch or potato starch together into a large bowl; set aside.
For the marshmallows:
Lightly coat an 8-by-8-inch baking pan with cooking spray; set aside.
Whisk together the gelatin and 1/2 cup of the cold water in a small microwave-safe bowl and let soften at room temperature for 5 minutes.
Stir together the sugar, 1/4 cup of the corn syrup, remaining 1/4 cup water, and salt in a medium saucepan over high heat. Bring to a boil, stirring occasionally, until the temperature reaches 240°F on a candy/fat thermometer. Meanwhile, pour the remaining 1/4 cup of the corn syrup into the bowl of a stand mixer fitted with a whisk attachment. Microwave the gelatin on high until completely melted, about 30 seconds. Pour it into the mixer bowl. Turn the mixer to low and keep it running.
When the syrup reaches 240°F, slowly pour it into the mixer bowl (the mixer should still be running). Increase the speed to medium and whip for 5 minutes. Increase the speed to medium high and whip 5 minutes more. Add the vanilla, increase the speed to the highest setting, and whip until opaque white, fluffy, and tripled in volume, about 1 to 2 minutes more.
Pour the marshmallow mixture into the prepared pan and use an offset spatula to smooth it into the corners. Sift 1/2 cup of the coating evenly and generously over top; set the remaining coating aside. Let the marshmallow set uncovered for at least 6 hours in a cool, dry place.
Use a knife to loosen the marshmallow slab from the edges of the pan. Dust a work surface with a thin layer of the reserved coating. Invert the marshmallow slab onto the work surface and dust it with more coating. Cut into whatever size pieces you wish (a pizza cutter works great for squares). Dip the sticky edges of the marshmallows in the remaining coating, patting off the excess. Store in an airtight container at room temperature for up to 1 month.
Variation: Increase the vanilla flavor by adding a scraped vanilla bean or a dab of pure vanilla bean paste along with the vanilla extract.