In a large bowl, stir together the butter and oil. Add in the sugar and stir until combined. Lightly beat the eggs, then stir them into the mixture with the peppermint extract.

In another bowl, combine the flour, baking powder, salt and baking soda. Stir 1/3 of the flour mixture into the wet ingredients, followed by half of the buttermilk. Repeat with another 1/3 of the flour, the remaining buttermilk, then the remaining flour, stirring just until combined after each addition. The batter will be quite thick - thicker than a cake batter, but thinner than a cookie dough.

Pour the batter into a large ziplock bag. Snip off the corner and pipe the batter into the donut pan, filling each cavity about 2/3 full. (My donut pan only makes 6 at a time, so I can only cook 6 at a time.)

Bake the donuts until they spring back when touched, 7-9 minutes. Let the donuts cool completely.

To make the glaze, stir together the powdered sugar, 3 tablespoons of cream and the vanilla. Add more cream if needed.

Place the crushed candy canes on a plate or in a shallow bowl. Dip the donuts into the glaze, then immediately into the candy canes.