Lightly grease a 9×13 inch baking pan. Mix the powdered sugar and cornstarch together in a small bowl. Dust the bottom and sides of the prepared pan generously with this mixture. Set aside.
Pour 1/2 cup of the cold water into the bowl of a stand mixer fitted with the whisk attachment. Sprinkle the gelatin over the water and let it bloom for 10 minutes.
In a medium saucepan, combine the granulated sugar, corn syrup, 1/2 cup water, and salt. Heat over medium heat, stirring until the sugar dissolves. Stop stirring once it boils.
Attach a candy thermometer to the side of the saucepan. Continue boiling the syrup without stirring until it reaches 240 degrees Fahrenheit (soft-ball stage). This usually takes about 8 to 10 minutes.
Once the syrup reaches 240 degrees F, carefully remove the pan from the heat. Turn the stand mixer on low speed. Slowly and carefully pour the hot syrup down the side of the mixing bowl into the gelatin mixture. Avoid pouring directly onto the whisk.
Increase the mixer speed to high. Beat the mixture for 12 to 15 minutes until the marshmallow is very thick, glossy, and has cooled significantly. It should hold stiff peaks.
Stop the mixer. Add the vanilla extract and mix briefly until just combined.
Scrape the marshmallow mixture into the prepared baking pan. Use a lightly oiled spatula to spread it evenly. Dust the top surface lightly with some of the reserved powdered sugar and cornstarch mixture.
Let the marshmallows set uncovered at room temperature for at least 4 hours, or preferably overnight.
Once set, turn the marshmallow out onto a cutting board dusted with the remaining powder mixture. Use a sharp, oiled knife or a pizza cutter to cut the slab into squares. Toss the cut sides in the powder mixture to prevent sticking.
Notes
For a gourmet flavor, substitute the vanilla extract with 1 vanilla bean, scraped, added to the syrup mixture before heating.
If you do not have corn syrup, you can substitute it with an equal amount of liquid glucose or honey, though the texture may change slightly.
To store, place the cut marshmallows in an airtight container separated by parchment paper for up to two weeks.