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 a candy thermometer reads exactly 240 degrees 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.
In a separate bowl, combine corn starch and powdered sugar. Spray 13x9 inch baking dish with cooking spray as well as a rubber spatula. Add about 1/3 of the sugar and corn starch blend, then shake the pan to evenly coat the bottom and sides. Use 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 corn starch 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 remaining sugar and corn starch mixture to coat all sticky sides.
Store in air tight container. They can also be frozen, if needed.
Notes
* I don't have a candy thermometer (yet), so I used my digital meat thermometer and it worked perfectly.