Preheat oven to 350°F.
Cream the butter and both sugars together in a large mixing bowl.
Mix in the coconut oil.
Beat in eggs and vanilla.
In a medium mixing bowl, whisk together the flour, cream of tarter, baking soda and salt.
Stir the dry ingredients into the wet.
Mix in lime zest.
Roll into 1 inch balls, roll them in sugar, and place on a cookie sheet.
Press with a glass or other smooth round object until flattened ⅓ inch thick.
Bake for 7-8 minutes or until just barely browned around the bottom.
Let cool on cookie sheet for 2 minutes before removing to a wire rack to cool completely.
To make the glaze, beat together the powdered sugar, lime juice, lime zest and enough of the milk to make a smooth consistency.
Spread on cooled cookies.
