Combine butter, oil, sugars and vanilla until smooth. Blend in eggs.
Add flour, salt, cream of tarter and baking soda until blended in.
Refrigerate dough 1+ hours.
Preheat oven to 350°F.
Use a small cookie scoop to make dough balls and press lightly with a glass.
Bake 8 minutes, watching carefully as to not over bake.
Allow to cool on cookie sheet.
Combine butter, powdered sugar, milk and peppermint extract until smooth. Ice cookies when completely cool. Optional: dip in crushed candy canes to garnish.
Notes
You can crush candy canes or peppermints or use crushed peppermint