In the large bowl of a stand mixer, combine the softened butter, granulated sugar, lime extract or key lime juice, and vanilla extract.
Beat the mixture on medium-high speed for approximately 2 minutes, or until it is light, fluffy, and pale yellow in color.
Add the eggs one at a time, beating well after each addition.
Add the freshly grated key lime zest and mix until evenly incorporated.
In a separate medium mixing bowl, whisk together the all-purpose flour, baking soda, and salt.
With the mixer set to low speed, gradually add the flour mixture to the butter mixture.
Mix only until the dry ingredients are fully moistened and a dough forms.
Stop the mixer and scrape down the sides of the bowl as needed to ensure even mixing.
Using a large spoon or rubber spatula, fold in the white chocolate chips and sweetened flaked coconut by hand.
Cover the cookie dough and refrigerate for 30 minutes to 1 hour.
Preheat the oven to 375°F.
Line two cookie sheets with parchment paper or silicone baking mats.
Using a medium ice cream scoop, portion the dough into mounds of approximately 1 1/2 tbsp each.
Place the dough mounds on the prepared cookie sheets, leaving approximately 3 inches between each cookie to allow for spreading.
Bake for 12 to 14 minutes, or until the cookies are lightly golden brown.
Remove the cookie sheets from the oven.
Allow the cookies to cool on the baking sheets for 3 minutes.
Transfer the cookies to a wire rack and allow them to cool completely.
Store the cooled cookies in a tightly sealed container at room temperature for up to 4 days.