To make the peppermint stick ice cream, heat the milk, salt, and sugar in a saucepan.
Set up an ice bath by placing a 2-quart (2l) bowl in a larger bowl partially filled with ice and water. Set a strainer over the top of the smaller bowl and pour the heavy cream into the bowl.
In a separate bowl, stir together the egg yolks. Rewarm the milk then gradually pour some of the milk into the yolks, whisking constantly as you pour. Scrape the warmed yolks and milk back into the saucepan. Cook over low heat, stirring constantly and scraping the bottom with a heat-resistant spatula, until the custard thickens enough to coat the spatula.
Strain the custard into the heavy cream. Stir over the ice until cool, add the mint extract and taste, adding more if desired. Whisk in the red food coloring – since brands vary, you can add enough until it’s the color you want. (I used 8 drops of McCormick red food coloring.) Refrigerate the mixture thoroughly, preferably overnight.
Freeze the custard in your ice cream maker according to the manufacturer’s instructions. Add the crushed candy canes to the ice cream during the last two minutes of churning. Scrape the churned peppermint stick custard into a chilled container and freeze until ready to serve.
To make the hot fudge sauce, melt the chocolate and butter in a bowl set over a pan of barely simmering water, stirring until smooth.
While the chocolate is melting, in a medium saucepan, whisk together the powdered sugar, brown sugar, cocoa powder, corn syrup (or golden syrup), salt, and heavy cream or crème fraîche. Heat the mixture, stirring frequently, until it comes to a low boil. Continue to cook, stirring constantly, until the sauce starts to thicken, about 45 seconds. Remove from heat and whisk in the melted chocolate.