In a mixing bowl or stand mixer ream together butter, vegetable oil, sugars, water, and eggs, zest of lime, and lime juice.
Mix until everything is combined.
In another bowl combine you dry ingredients. Now slowly mix the dry ingredients into the wet.
Your dough should be crumbly, not sticky.
Roll a golf ball sized ball of dough(50g) and place it on your cookie sheet.
Put 1/4 c of sugar and a pinch of salt in a dish (this is in addition to the sugar and salt listed above.) Stick the bottom of a glass in it. This is going to be your cookie press.
Firmly press it into the center of your dough ball. You want your dough to spill out over the sides of the glass.
Bake at 350 degrees F for 10-12 minutes. Don't overcook them, they should just barely be browning on the bottom. Let sit for 1-2 minutes and then move cookies to a cooling rack. Let them cool completely before frosting.
Start by creaming together butter, sour cream, and coconut extract. Slowly add powdered sugar until it's thick and fluffy. (Depending on your climate you may need more or less than 3 cups. Just keep adding until it's thick enough to spread onto the cookies. You don't want it to be runny and dripping off your cookies)
Frost cookies and place back back on a cookie sheet. Place them in the freezer (do not cover them) for 10-15 minutes or until the frosting is firm. Now start layering them in a large storage container using parchment paper between layers so they don't stick. Keep them in the freezer until you're ready to serve them.
The magic of these cookies is that they are served cold. Take one out of the freezer 15 minutes before you plan to eat it and it will be perfect!
Notes
Search MyFitnessPal for Stay Fit Mom Coconut Lime Sugar Cookie to log your cookie per gram!