Coat a 9-inch square pan with cooking spray. Line bottom of pan with parchment paper and coat with cooking spray; set aside.
In a small saucepan, combine sugar, corn syrup, and 3/4 cups water. Set to medium heat and cook, stirring until sugar is dissolved. Stop stirring and increase heat to medium-high. Bring mixture to a boil and cook until it registers 260 degrees on a candy thermometer.
Meanwhile, pour an additional 3/4 cups water into a heatproof bowl. Stir in gelatin and let stand 5 minutes. Set bowl over a small pan of simmering water; whisk constantly with a fork until gelatin is dissolved into mixture (it may be lumpy at first – just keep whisking). Remove from heat. Stir in peppermint extract and set aside.
Beat egg whites in the bowl of an electric mixer fitted with the whisk attachment until stiff, glossy peaks form.
Meanwhile, whisk gelatin mixture into sugar mixture. With the mixer running, gradually add gelatin-sugar mixture to egg whites. Mix on highest speed until very thick, 10 to 12 minutes.
Pour marshmallow mixture into prepared pan (you may end up with excess). Working quickly (marshmallow will begin to set), drop dots of red food coloring across surface. Using a wooden skewer, swirl food coloring into marshmallow to create a marbleized effect. Let marshmallow stand, uncovered, at room temperature until firm, at least 3 hours or overnight. Cut into squares.
Package marshmallows as gifts with hot cocoa mix, or keep for yourself to enjoy!
Store in an airtight container at room temperature for up to 1 week.