Grind enough candy canes up in blender or magic bullet to make 1/4 cup.
Whisk cream cheese in bowl until creamy.
Set a large bowl with ice that this bowl will fit into to cool.
Mix 2 tablespoons of milk with cornstarch to make a slurry.
Cook remaining milk, cream, sugar corn syrup in heavy bottomed 4 quart pot, including one candy cane. Bring to rolling boil and boil for 4 minutes on medium heat.
Remove from heat. Whisk in slurry of corn starch. Boil again until thickened, over med. high heat, about a minute.
Pour into bowl that has the cream cheese. Place that bowl in ice bath. Cool.
Place in refrigerator for 24 hours.
Use your ice cream maker.
Add in peppermint extract, peppermint schnapps and crushed candy canes during the last five minutes of ice cream churning.
Put in plastic with lid. Cover first with parchment. Freeze for at least 4 hours before serving.