Pink Peppermint Stick Ice Cream for Christmas- or Afterwards

Years go I got one of those Cuisineart Ice Cream Machines that uses a frozen container that spins in the machine to make ice cream. The instructions were always poor and I put the machine in the appliance storage area, to rarely be used. Of course there is a good reason not to have an ice cream machine near by, but Christmas and the season of breaking your diet is the only time it might come out.
For this Christmas I decided I would try one more time to make ice cream with the machine and if it did not turn out I could make something else. My father always liked peppermint stick ice cream with chocolate sauce from a restaurant called the Rice Patty in Georgetown, SC, so I attempted to recreate that.
Since I have never had any luck with the recipes that came with the machine I read number of recipes online and made a mashup of various ones. Carter declared it was the best ice cream ever.  
1 cup milk

2 cups heavy cream

2 eggs

1/3 cup of sugar

1 small package of sugar free instant vanilla pudding (the real secret)

1/3 cup of crushed up candy canes

Good splash of peppermint extract

Pinch of salt
In a kitchen aid stand mixer put the milk, cream, eggs and sugar and whisk on medium high until it gets a little thicker.
Add the vanilla pudding mix and beat again for 20 seconds.
Add the peppermint candy and extract and salt and mix again.
Pour the mixture into a container with a lid and chill for at least four hours, but overnight is better.
Make sure the freezer bowl section of the ice cram maker has been in the freezer for at least six hours. Pour the ice cream liquid into it and place the blade in and put in the ice cream machine. Let spin for at least 20 minutes or until the it is frozen.
Remove the ice cream into a container and put in the freezer to finish hardening up. The best thing about it is that the pudding keeps it from getting too hard.
Serve with fudge sauce and crushed peppermint stick on top.