Preheat oven to 350 degrees. Line 2 baking sheets with parchment paper or baking mats, set aside.
In the bowl of a stand mixer, add butter, sugar, salt and coconut extract. Turn mixing speed on low, then gradually turn it up to medium. Mix just until combined, about 1 minute. With mixing speed on low, gradually add the flour, mix just until combined.
Add egg to a small mixing bowl and whisk. Add coconut to a medium mixing bowl.
Using a tablespoon, scoop the dough one at a time and shape into balls. Roll each dough ball in the egg, then in coconut, press to help the coconut stick. Transfer cookies to prepared sheets, placing them 2 inches apart. Using your thumb, make an indentation in the center of each ball of dough. Transfer the baking sheet to freezer and for 10 minutes.
Place in oven and bake for 11 minutes, then remove from the oven and re-indent cookies. Return the cookies to the oven and bake for 8 minutes longer or until the coconut is golden brown and cookies are set. Remove from the oven and let cool on the sheets for 5 minutes before transferring to a wire rack to cool completely.
To make the caramel topping: put a small bowl of cold water and a pastry brush (preferably silicone) near the stovetop.
Put the sugar, water and corn syrup in a small saucepan or skillet and bring to a boil over medium-high heat. Allow the ingredients to bubble away, brushing down the sides of the pan with cold water if there are spatters and swirling the pan a couple of times once you start seeing some color, until the caramel turns a pale to medium amber (this generally takes about 8-10 minutes). Lower the heat, stand away from the pan and add the cream, followed by the butter and salt — ingredients that might cause the caramel to roil. Remove the pan from the heat, drop in the chocolate and stir to combine.
Fill each cookie with the caramel sauce, then add the chocolate eggs. Let the cookies set for 20 minutes before serving or storing.