You'll need a stand mixer with the whisk attachment, a 9x13-inch baking dish, and a candy thermometer or digital instant-read thermometer. Check the thermometer in boiling water first (see note 1).
For the marshmallow
In the bowl of a stand mixer, combine the gelatin with the ice cold water and give it a swirl to ensure the gelatin doesn't clump. Let sit.
In a small saucepan, combine sugar, corn syrup, salt, and water. Heat on medium high with the lid on for a few minutes to bring it to a boil. Then, remove the lid and heat on high heat until the thermometer reads exactly 240°F. This can take 8-10 minutes. As soon as it reaches temperature, remove it from the heat.
With the whisk attached to the stand mixer, start mixing the gelatin and water on low speed. Slowly pour the hot sugar mixture into the bowl. Once all of the hot sugar mixture has been added, turn the speed to high and continue to mix until you get a beautiful white marshmallow mixture that is just warm to the touch, about 13-14 minutes. With the mixer still running, add the vanilla and peppermint extracts and mix for one more minute.
To set and cut
In a separate bowl, combine cornstarch and powdered sugar. Spray a 9x13-inch baking dish and a rubber spatula with nonstick spray. Add about 1/3 of the sugar and cornstarch blend, then shake the pan to evenly coat the bottom and sides. Use the spatula to transfer the marshmallow mixture from the bowl to the baking dish and spread into an even layer. Top with half of the remaining sugar and cornstarch mixture, and spread it across the top to cover. Allow the marshmallows to sit uncovered 4-10 hours.
Lift the marshmallows out of the pan and transfer to a cutting board. Cut into desired shapes and then roll each individual marshmallow in the remaining sugar and cornstarch mixture to coat all sticky sides.
Notes
Thermometer: a candy thermometer or a digital instant-read thermometer both work. Check it in boiling water first; at sea level it should read 212°F. If it reads differently, move the 240°F target by the same amount. This also adjusts for altitude.
Storage: store in an airtight container at room temperature. For longer storage, freeze them; they stay soft in the freezer.
Yield: about 100 one-inch marshmallows from a 9x13-inch pan. Cut bigger squares to top hot chocolate.