In a large mixing bowl, beat together softened butter, margarine and both sugar. Gradually add the oil, beating until smooth, then beat in the egg and the extracts.
Mix together the flour, salt, baking soda and cream of tartar and add to the butter mixture. Stir until you have a smooth dough.
Drop 16 quarter cup size (approximately rounds of dough onto two plates lined with plastic wrap. Press rounds down slightly, then cover and chill for an hour or until ready to bake.
Preheat the oven to 350 degrees F.
Arrange 8 mounds to a baking sheet and bake for about 12 to 15 minutes or until edges just start to brown.
To make the frosting, beat the softened cream cheese and butter until blended, then beat in the extract. Gradually add the sugar, beating and scraping bowl, until frosting is creamy and smooth. Add a little bit of red food coloring and beat until pink. Spread over the cookies.