In a medium saucepan placed over medium heat, add the half-and-half and 1 ½ cups of the heavy cream. Add sugar and salt and whisk until dissolved.
While the cream mixture heats, add 2 egg yolks to the remaining ½ cup of heavy cream, whisking to combine. When the cream is hot, slowly ladle some of the hot cream into the cream and egg yolks, whisking continuously. This tempers the eggs.
Then slowly pour the egg yolk mixture into the hot cream mixture, while whisking. Cook over medium-low heat until slightly thickened, being careful not to boil the cream mixture.
Strain the mixture into a heat-proof bowl. Stir in peppermint extract and red food coloring. The amount of food coloring can be adjusted to get the shade of pink desired. Cover with plastic wrap, pressing the plastic wrap directly on top of the cream mixture. Refrigerate until chilled, at least 2 hours.
Meanwhile, add the Oreo cookies to the bowl of a food processor. Pulse until the cookies resemble fine crumbs. Drizzle in the melted butter and pulse to combine.
Pour the crumbs into a 9-inch springform pan. Using the bottom of a glass, press the crumbs evenly over the bottom and about half-way up the sides. Put the pan in the freezer to firm up.
When the cream mixture is chilled, transfer it to an ice cream machine and churn until the mixture is the consistency of soft serve. Mix in the crushed candy canes. Pour the mixture into the Oreo crust and freeze until firm, at least 4 hours but preferably overnight.
Add the heavy cream to the bowl of a stand mixer fitted with a whisk attachment. Whisk until soft peaks form. Add the powdered sugar and extract. Whisk until still peaks form.
Transfer the whipped cream to a piping bag fitted with a tip. (I used a 2D tip.) Decorate the top of the pie as desired and sprinkle with crushed candy canes. Put back in the freezer to set the whipped cream. Slice and serve.