Preheat oven to 350° F. Line a baking sheet with parchment paper. Set aside.
In the bowl of a stand mixer with the paddle attachment, combine the butter, granulated sugar, and confectioners' sugar. Mix until just combined.
Drizzle in oil and pour in eggs, water, and vanilla extract. Mix well.
In another bowl, whisk together flour, cocoa, salt, baking soda, and cream of tartar.
Slowly add the flour mixture to the butter mixture and allow the mixer to combine until the dough pulls away from the sides of the mixer bowl.
Fold in the coconut.
Using a 2 tablespoon scoop, drop golf-ball-size portions of the dough on the lined cookie sheet.
Roll out each cookie dough ball until it is smooth and round.
Using a kitchen glass that has been pressed into granulated sugar, press down on each cookie dough ball until the center of the cookie is pressed flatter than the edges. Place them on a baking sheet about 2 inches apart. Repeat this on all cookies.
Bake for 7-8 minutes or until edges are set; tops may still look a little wet. Let the cookies cool before adding the topping.