Place Ice Cream Maker Freezer “Bowl” in freezer at least 12 hours before using.
Combine heavy cream, 1 cup milk, and salt in a medium saucepan over medium heat. Bring to a simmer.
In a small bowl, combine 1 1/2 tablespoons milk and cornstarch; stir thoroughly until there are no lumps. Add to the simmering cream mixture and cook for 5 minutes (at a simmer or light boil). (This whole step with the cornstarch is optional).
Meanwhile, in a medium mixing bowl, beat the sugar and corn syrup into the egg yolks.
When the cream mixture is warm, slowly pour about 1/3 to 1/2 of the cream mixture into the sugar & egg yolk mixture, stirring frequently while you are pouring (stirring keeps the eggs from scrambling). Transfer all of the mixture from the bowl back into the medium saucepan. Turn heat to low. Stir frequently with a wooden spoon, for about 5 minutes, until custard thickens slightly (you do not want to the mixture to get too hot, or the eggs will scramble). Transfer the mixture to a medium mixing bowl. Stir in peppermint extract and vanilla extract. Cover with plastic wrap. Transfer to refrigerator to cool for at least 4-5 hours (can accelerate process by placing in freezer for 1 hour and then transferring to refrigerator). Can leave in refrigerator 1-2 days, if necessary.
Before you are ready to make the ice cream, transfer the mixing bowl with the ice cream mixture to the freezer for 30-40 minutes, stirring every 15-20 minutes in order to keep the edges from freezing (you don’t actually want the mixture to freeze, just to get colder before putting it in the machine).
Transfer the mixture to Ice Cream Maker Freezer “Bowl.” Place the bowl in the machine and churn the ice cream. It should take approximately 25 to 30 minutes.
Transfer the ice cream to the container(s) you are going to store it in. Working as quickly as you can, stir in some of the ginger snap pieces and candy cane pieces. Fold over and mix with a spoon. Repeat until almost all of the ginger snap pieces and candy cane pieces are folded in. Smooth out the top. Sprinkle with remaining ginger snap pieces and candy cane pieces. Place in the freezer (preferably, put the container directly in the ice bin, if you have one, nestled in the ice, so that it will cool as quickly as possible). Freeze for at least 5 hours.
Before serving, let ice cream rest for 5-10 minutes at room temperature, where it should just being to start to melt (you don’t want to eat ice cream straight from the freezer—it is too hard and to cold—you want it to warm up a little bit so it is slightly softened).
* To get crushed ginger snaps, can just chop up ginger cookies with a knife. Chop it so that there is a mixture of sizes (some almost just crumbled cookies and then some actually nice-sized pieces, 1/2-inch to 3/4-inch).
** To crush candy canes, place unwrapped candy canes in two supermarket plastic bags. Using a meat mallet or rolling pin or other heavy object, crush the candy canes. Once again, you want different size pieces, although you don’t want any of the candy cane pieces to be too big (no bigger than about 3/8-inch). In general, the candy cane pieces will be much smaller than the ginger snap pieces. After crushing some of the candy canes, you may need to add another plastic bag on the outside of the other bags (crushing the candy canes with the mallet can make holes in the existing plastic bags).