Line a 9-by-13-inch baking dish with parchment paper letting the paper overhang on two sides. Dust the paper and pan liberally with the confectioners' sugar.
Put 1/2 cup cool water in the bowl of a stand mixer fitted with a whisk attachment and sprinkle the gelatin over the water.
Combine the granulated sugar, corn syrup, salt and 1/2 cup water in a medium saucepan. Place over medium-high heat and cook, swirling the pan occasionally, until the sugar mixture reaches 240 degrees F on a candy thermometer, 12 to 15 minutes.
With the mixer on medium-low speed, very slowly pour the hot sugar syrup into the gelatin mixture. Continue to beat until foamy, about 1 minute, then increase the speed to high and beat until the mixture forms stiff peaks and the bottom of the bowl is just slightly warm to the touch, 10 to 15 minutes.
Add the peppermint extract and the pink food coloring and mix again to combine. Spread the marshmallow mixture into the prepared pan and smooth the top with an offset spatula sprayed with cooking spray. Sprinkle the top with the crushed candy. Let sit, uncovered, overnight.
Lift the marshmallow out of the pan using the overhanging parchment as handles and place on a cutting board. Use a sharp, heavy knife sprayed with cooking spray to trim the edges. Cut the marshmallow into 12 pieces, spraying the knife between cuts as needed.
To store the marshmallows, dust the bottom of an airtight container with confectioners' sugar. Remove the marshmallows from the parchment, place in the container and keep at room temperature for up to a week.
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