Preheat ovedn to 350 degrees F. Line 2 baking sheets with parchment paper.
Whisk together flour, coconut, baking soda, baking powder, and salt. Set aside
Put 1 1/2 C sugar, 1 tsp. lime zest and cream cheese in a bowl.
Place remaining 1/3 C sugar in a flat dish or pie plate and set aside.
Pour warm butter over sugar and cream cheese and mix together (there will be some small lumps)
Whisk in oil.
Add egg, milk, lime juice and mix until smooth.
Add flour mixture and mix until dough forms.
Divide dough into 24 equal pieces and roll each into a ball.
Roll balls in reserved sugar to coat and place on baking sheet.
Use bottom of drinking glass to flatten dough balls until they are about 2 inches in diameter.
Sprinkle tops evenly with sugar remaining in shallow dish.
Bake one tray at a time for 11-13 minutes or until edges are set and just beginning to brown.
Cool on baking sheet for 5 minutes and then transfer to wire rack.
Mix together powdered sugar, lime zest and milk until a nice spreading consistency.
Frost cooled cookies.