Preheat the oven to 350 degrees.
In a stand mixer bowl, cream together your softened butter, 1 1/4 cup white sugar and powdered sugar for about 2 minutes until smooth.
Add in the coconut oil, lime juice, lime zest and eggs. Mix well.
Add in the flour, baking soda, cream of tartar and salt. Mix well until everything comes together, into a slightly crumbly dough.
Scoop the cookie dough into 3 TBS sizes balls and roll into balls.
Place on a baking sheet lines with a silicone liner or parchment paper. (No more than 8 cookies per tray.)
Add the remaining 1/4 cup sugar to a small bowl.
Spray the bottom of a flat circular glass cup with nonstick spray. Stick it into the sugar.
Press each cookie down using the glass cup until they're about 1/2 inch thick.
Bake in the preheated oven for 7-10 minutes until the cookies look barely done. They may not look cooked through in the middle.
Allow cookies to cool for several minutes on the baking pan. Then remove them to a cooling rack to cool completely.