Preheat oven to 350 degrees and very lightly grease a baking sheet.
In a large bowl, cream together butter, vegetable oil, 1 1/4 cups sugar, powdered sugar, heavy cream, eggs, and peppermint extract.
In a second large bowl, whisk together flour, baking soda, cream of tartar, and salt.
Add dry ingredients to wet ingredients and mix until dough comes together.
Roll dough into balls (roughly the size of golf balls) and place on greased baking sheet 2-3 inches apart.
Place remaining 1/4 cup sugar in a shallow dish. Press the bottom of a tall drinking glass into the cookie dough in the bowl (just to moisten the bottom of the glass, no cookie dough should stick to it), then dip into sugar. Press the bottom of the glass into the top of each cookie dough ball to flatten about halfway. (The edge should "break" to create the crinkle edge")
Bake for 8-11 minutes until cookies look dry but NOT browned at all. Allow to cool on baking sheet for 4-5 minutes, then transfer to a cooling rack to cool completely. Repeat process with remaining cookie dough.
For the frosting, cream butter and cream cheese until very light and fluffy (about 2-3 minutes). Stir in peppermint and vanilla extracts. Mix in powdered sugar until incorporated and smooth. Add 2-3 drops of red food coloring and mix well until every distributed.
Store cookies in airtight container at room temperature and frosting in airtight container in the fridge. Frost cookies immediately before serving and sprinkle with crushed candy canes.