2 cups heavy whipping cream
First make the custard. In a large bowl, whisk together the eggs, sugar and salt. Set aside. Combine the cream and milk in a heavy saucepan. Split the vanilla bean lengthwise and scrape the seeds into the cream. Add the entire bean as well. Gently heat over medium heat until very hot. Do not let it boil. Remove the vanilla bean and then gradually pour the hot cream into the egg mixture, whisking furiously, to temper the eggs, until all the cream and egg-mixture are combined.
Return the custard to the saucepan and gradually heat again over medium heat, stirring often and until the custard is thickened and coats the back of a wooden spoon. Then, pour the custard into a clean bowl through a fine mesh sieve (to get out any lumpy bits). Stir in peppermint extract. Cover tightly with plastic wrap (pushing the plastic wrap down to also cover the surface of the custard) and cool in the fridge for at least 2 hours or overnight.
Add the cooled custard to an ice cream maker and freeze according to the manufacturer's instructions, adding half the crushed candy canes about 10 minutes into the freezing cycle (the more you add the earlier in the cycle the pinker the ice cream so you can adjust timing and amount to get the pinkness you want). Fold the remaining candy canes into the ice cream just before putting into the container in the freezer. Freeze and enjoy when frosty!