In a bowl, beat together the butter, oil, and both sugars.
Add eggs, vanilla, and orange zest. Beat in until well combined.
Add in baking soda, cream of tartar, and flour. Beat until just combined.
Using a cookie scoop, scoop dough onto an ungreased baking sheet. Bake at 375 for 8 to 10 minutes until lightly browned. Cool on wire racks.
In a bowl, whisk all of the icing ingredients together. You want it spreadable and not runny. Add more sugar if needed. Spread on cooled cookies. Makes about 5 dozen.