Preheat oven to 350°F and line baking sheet.
Beat butter, then mix in oil, sugars, and spices.
Add egg and vanilla and beat until combined.
Mix in flour, baking soda, salt.
Chill dough for 1 hour or freeze for 15 minutes.
Roll dough into balls, coat with cinnamon sugar.
Bake for 10–14 minutes depending on size.
Cool 10 minutes on tray before transferring.
Whisk glaze and spread on cooled cookies.
Sprinkle with nutmeg.
