These lightly baked dome cookies provide a double-dose of tropical flavor -- delicate coconut macaroons sandwich a creamy coconut filling.
Stir together coconut, granulated sugar, egg whites, coconut extract, and salt in a large bowl. Refrigerate, covered, until cold, at least 1 hour or overnight.
Preheat oven to 325 degrees. Form heaping teaspoons of dough into balls; space 1 1/2 inches apart on baking sheets lined with parchment paper. Gently flatten to about 1 1/2 inches in diameter. Bake cookies, rotating sheets halfway through, until edges begin to turn golden, 9 to 10 minutes. Transfer cookies to wire racks; let cool completely.
Put butter, cream of coconut, and shortening in the bowl of an electric mixer fitted with the paddle attachment. Mix on medium speed until smooth, about 1 minute. Add confectioners sugar and coconut extract; mix until pale and fluffy, about 2 minutes.
Place a heaping teaspoon of filling on the bottom of one cookie. Sandwich with another cookie. Repeat with remaining cookies and filling. Transfer to a platter; cover with plastic wrap. Refrigerate until filling is firm, about 30 minutes. Let stand at room temperature 10 minutes before serving. Cookies can be stored between layers of parchment in airtight containers at room temperature up to 3 days.