Line a 9x13-inch pan with foil, enough so that excess foil hangs over the sides of the pan. Spray with non-stick cooking spray; set aside.
In a small bowl, whisk together the powdered sugar and cornstarch; set aside.
Pour ½ cup of the cold water into the bowl of an electric mixer fitted with the whisk attachment. Sprinkle the gelatin over the water. Let stand until the gelatin becomes very firm, about 15 minutes.
Meanwhile, combine the remaining water and the corn syrup in a medium saucepan. Pour the sugar into the center of the saucepan and add the salt. Place the pan over medium-high heat and bring to a boil, gently swirling the pan, until the sugar has dissolved completely and the mixture reaches 240 degrees F, about 6 to 8 minutes.
Turn the mixer on low speed and carefully pour the hot syrup mixture into the gelatin mixture, avoiding the side of the bowl as much as possible. Gradually increase the speed to high and whip until the mixture is very thick and stiff, 10 to 12 minutes, scraping down the bowl as needed. Add the vanilla and peppermint extract and mix until incorporated, about 15 seconds.
Working quickly, scrape the mixture into the prepared pan using a spatula that has been sprayed with non-stick cooking spray. Smooth the top into an even layer. Sift 2 tablespoons of the powdered sugar mixture over the pan. Cover and let sit overnight at room temperature.
The next day, turn the marshmallow slab out onto a cutting board and peel off the foil. Sift 2 tablespoons of the powdered sugar mixture over the slab. Using a pizza cutter or a sharp knife that has been sprayed with non-stick cooking spray, cut into 1-inch strips one way, and then across the other way for square marshmallows. You could also use round cutters to cut them out as well.
Place the remaining powdered sugar mixture in a large ziploc bag. Working with 3 or 4 marshmallows at a time, toss the marshmallows in the bag with the powdered sugar mixture, then toss in a fine-mesh strainer to remove any excess powder. Marshmallows can be stored in an airtight container or bag for up to 2 weeks.