Place 1/2 cup of the cold water in the bowl of stand mixer. Sprinkle gelatin over water and let sit (gelatin will absorb water and soften, this is called "blooming").
Attach wire whisk attachment to the mixer.
Combine remaining 1/2 cup water in a saucepan with granulated sugar, corn syrup, and salt. Set over medium heat and bring to a boil, stirring to dissolve sugar.
Once mixture comes to a boil, insert a candy thermometer and do not stir anymore. Use a wet pastry brush to rub and dissolve any granulated sugar from the sides of the pan.
Cook sugar mixture to 240°F. It will come to 220°F pretty quickly, and then take about 5-8 minutes to increase to 240°F. Keep a close eye, still no stirring.
As soon as sugar is at 240°F remove from heat and turn stand mixer on low speed. Pour the hot sugar into the bowl of the mixer in a slow steady stream, aiming for the side of the bowl and not on top of the whisk.
Once all the hot sugar has been added, increase speed to medium high and beat until mixture is white and thick and bowl is no longer hot to the touch. This will take about 5 minutes but will depend on exact speed and power of mixer. Add vanilla during the last few minutes of mixing, once marshmallow starts to thicken.
How to know if marshmallow is done: Stop mixer, lift whisk and watch how marshmallow mixture falls back into bowl. It should hold its shape for a few moments and then absorb back into the rest of the mixture. (If marshmallow flows quickly off whisk and immediately absorbs back in then you need to beat longer, if marshmallow is stiff and doesn't flow off whisk without scraping then it is over beaten).
Immediately scrape mixture out of bowl with a spatula into prepared pan. Use spatula to gently spread it into an even layer.
Let marshmallows dry uncovered for at least 4 hours before slicing.