Because they use modernist ingredients, these amounts are given in metric by weight. For more information on how to measure modernist ingredients check out this article.
Attach the whisk attachment to your standing mixer. Add the ingredients for the gelatin base and let the gelatin hydrate for 5 to 10 minutes.
Prepare a 13" x 9" baking pan by spraying it with Pam. Combine the cornstarch and powdered sugar and sift some of the cornstarch mixture in the pan.
While the gelatin is blooming, combine the ingredients for the syrup in a sauce pan set over medium-high to high heat. Stir it lightly until the sugar is dissolved and then leave it alone while it heats. Cook the syrup until the temperature reaches 230°F to 240°F (110°C to 116°C).
Remove the syrup from the heat and pour it into the standing mixer on top of the bloomed gelatin and water. Turn the mixer on low and slowly increase the speed until it is on high, being careful not to splash the hot syrup out of the mixer.
Whip the marshmallow mixture until it has tripled in volume, it should take 8 to 15 minutes. When the marshmallows have almost been fully whipped, add the vanilla extract and make sure it is fully incorporated.
Pour the marshmallow mixture into the pan and spread it out evenly. Sift the top with some of the cornstarch mixture. Let the marshmallow set for several hours, preferably overnight.
Dust a cutting board with the cornstarch mixture. Turn the marshmallows out onto the cutting board and cut into the shapes you want. Coat the shapes with the cornstarch mixture to prevent them from sticking to each other.
I've found the best way to cut the marshmallows into squares is using a lightly oiled rolling pizza cutter, which I first heard suggested by Alton Brown. However, using a lightly oiled knife works well, as do kitchen shears. You can also use lightly oiled cookie cutters for unique shapes.
You can store the marshmallows for several days in a sealed container.