Coconut-Lime Layer Cake

Cake

1. Step 1Preheat oven to 350°. Grease 3 (8") baking pans with cooking spray. Line bottoms of pans with parchment, then grease with cooking spray. In a medium bowl, whisk flour, baking powder, salt, and baking soda. 
2. Step 2In a large bowl, using a handheld mixer (or in the large bowl of a stand mixer fitted with the whisk attachment), beat granulated sugar, butter, lime zest, and vanilla on medium-high speed until light and fluffy, about 3 minutes. Beat in cream of coconut until well combined. Add eggs, one at a time, beating well after each addition. Add half dry ingredients and beat on low speed until just combined. Add buttermilk and remaining dry ingredients and beat on low until just combined. 
3. Step 3Divide batter among prepared pans; smooth tops. Bake cakes, rotating pans halfway through, until a tester inserted into the center comes out clean, 35 to 45 minutes.
4. Step 4Transfer cakes to wire racks and let cool in pans 10 minutes. (Leave oven on.) Invert cakes onto racks. Remove parchment and let cakes cool completely.

Curd

1. Step 1In a medium saucepan, whisk egg yolks and granulated sugar until well combined and lightened in color. Whisk in lime juice and salt. Add butter. Cook over medium heat, whisking constantly, until butter melts and mixture is thick enough to coat the back of a spoon, about 5 minutes. 
2. Step 2Transfer lime curd to a medium bowl and press a piece of plastic wrap directly on surface to prevent a skin from forming. Refrigerate until well chilled, about 1 hour.

Frosting

In a large bowl, using a handheld mixer (or in the large bowl of a stand mixer fitted with the whisk attachment), beat powdered sugar, butter, cream cheese, coconut extract (if using), and salt on medium-high speed until light and fluffy, about 3 minutes. Wrap bowl in plastic and store at room temperature until ready use.

Assembly

1. Step 1Invert cakes so they are domed side up. Using a serrated knife, carefully cut off rounded tops to level cake layers. Transfer about 1/2 cup frosting to a pastry bag or zip-top bag. Cut about 1/4" off bottom of pastry bag or one bottom corner of zip-top bag.
2. Step 2Arrange one cake cut side up on a platter. Pipe a 1/4"-wide circle of frosting just inside edge of cake layer (this frosting will create a dam, preventing the curd from squishing out). Spoon and spread half of lime curd inside frosting circle. Top with another cake layer cut side up. Repeat frosting and filling process. Top with third cake layer cut side down. Refrigerate cake until curd and frosting set, about 1 hour.
3. Step 3Meanwhile, spread coconut flakes on a rimmed baking sheet. Bake, stirring occasionally, until light golden, about 10 minutes. Transfer to a wire rack and let cool completely on baking sheet.
4. Step 4Transfer 3/4 cup frosting to a pastry bag fitted with a 1/4" star tip or spoon into a zip-top bag (if using a zip-top bag, push frosting into one corner of bag, then snip a bit off the corner). Frost sides and top of cake with remaining frosting in bowl, then press coconut into sides and top of cake.
5. Step 5Pipe swirls of frosting on top of cake. Place a lime quarter in each swirl. Refrigerate until set, about 1 hour.