First, make the chocolate cake by prepping your oven and cake pan. Position a rack in the center of the oven and preheat the oven to 350°F. Generously spray an 8-inch square pan with cooking spray and line the bottom with parchment paper. Spray the parchment, too. Mix the dry ingredients. In a medium bowl, whisk together the flour, coconut, baking powder, baking soda, and salt.
Whisk the eggs, then add the sugar, coconut milk, and coconut oil. In a large bowl, whisk the eggs until combined. Whisk in the sugar until thick and homogenous, then whisk in the coconut milk and coconut oil until combined.
Mix the dry ingredients into the wet ingredients. Sprinkle the dry ingredients over the wet ingredients all at once, and mix with a rubber spatula until combined.
Assemble the cake. Pour the batter into the prepared pan and use an offset spatula to smooth its top if necessary.
Bake the cake. Bake for 30 to 35 minutes. When done, the top of the cake should bounce back when gently pressed and a skewer inserted into the center of the cake should come out with a few crumbs attached. Cool completely in the pans on a wire rack before frosting.
Fill the cake. Once the cake is cool, Run a butter knife or offset spatula along the edges of the pan and use the overhanging parchment as handles to lift the cake out of the pan and onto a cutting board or wire rack.Use a wooden skewer or a fork to poke holes all over the top of the cake.Fill the pastry bag with the lemon curd. Pipe lemon curd into the holes of the cake, using an offset spatula to smooth out any extra curd over the top of the cake. Make the frosting. In the bowl of a stand mixer fitted with the paddle attachment, combine the confectioners’ sugar, butter, and salt for the frosting. Beat on medium-low until soft and creamy, 1 to 2 minutes. Scrape down the sides of the bowl with a rubber spatula.With the mixer on low, add 1 Tablespoon of the coconut milk for the frosting. Beat until combined. Scrape down the bottom and sides of the bowl once more, then beat on medium-high until the frosting is creamy and smooth, 2 to 3 minutes. Frost and decorate the coconut cake. Use an offset spatula to cover the top of the cake with the frosting completely. Garnish with extra unsweetened shredded coconut.
Serve and store. Serve at room temperature. The cake will keep in an airtight container for 1 day. After that, transfer to the refrigerator and refrigerate for up to 2 more days.