Prepare the ice cream: Let the ice cream soften slightly at room temperature, 5 minutes. Transfer to a stand mixer fitted with the paddle attachment and beat until creamy but not melted, about 1 minute. Add 1/2 teaspoon peppermint extract and the crushed candy canes and beat until just incorporated, about 30 seconds. Transfer to a small metal baking dish or loaf pan, cover and freeze until firm, 1 to 2 hours.
Meanwhile, preheat the oven to 300 degrees F and line a baking sheet with parchment paper. Using a bowl or cake pan as a guide, trace an 8-inch circle onto the parchment with a pencil, then flip the parchment over (the circle should still be visible); set aside.
Make the meringue: Fill a medium saucepan with 1 to 2 inches of water and bring to a simmer over medium-high heat; remove from the heat. Combine the egg whites, sugar and salt in a large heatproof bowl and set over the hot water (do not let the bowl touch the water). Whisk until the egg whites are foamy and hot, about 5 minutes. Remove the bowl from the pan and beat with a mixer on medium speed until stiff and shiny, about 10 minutes. Beat in the remaining 1/2 teaspoon peppermint extract until combined.
Transfer the meringue to a pastry bag fitted with a 1/2-inch round tip. Pipe in a tight spiral onto the prepared parchment-lined baking sheet, starting in the middle of the circle and working toward the outer edge. Pipe more meringue around the outer edge of the circle to build up the side about 2 inches high. Dip a small paintbrush or toothpick in the food coloring, then drag it around the meringue spiral to create streaks. Repeat several times.
Bake until the meringue is just firm but not brown, about 1 hour. (If it starts browning, reduce the oven temperature to 275 degrees F.) Transfer to a rack and let cool completely. (The meringue will continue to harden as it cools.) Carefully invert the meringue onto a platter and gently peel off the parchment. Invert again and set on the platter. Top with scoops of the prepared peppermint ice cream and sprinkle with more crushed candy canes. Freeze until ready to serve, up to 2 days.