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Preheat the oven to 350°F with a rack positioned in the bottom third. Butter two 9” x 2” round cake pans and dust them with flour, shaking out any excess.
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To make the cake: Sift the flour, baking powder, baking soda, cardamom, and salt into a large bowl; set aside.
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In a large measuring cup or small bowl, stir together the cream of coconut, water, and vanilla; set aside.
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In the bowl of a stand mixer fitted with the flat beater (or in a large mixing bowl using a handheld mixer), cream the butter and sugar together on medium-high speed until light and fluffy, 3 to 5 minutes.
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Add the eggs one at a time, beating well after each addition and scraping down the sides and bottom of the bowl with a rubber spatula as necessary.
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With the mixer on low speed add the flour mixture in thirds, alternating with the cream of coconut mixture and beginning and ending with the flour. Mix until just combined, scraping down the sides and bottom of the bowl as necessary.
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Divide the batter evenly between the prepared pans (about 830g per pan) and smooth the tops with a spatula. Tap the pans firmly on the counter to remove any air bubbles.
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To bake the cake: Bake the cake layers for 35 to 40 minutes, until a toothpick or thin paring knife inserted in the center of a layer comes out clean.
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Let the cakes cool in the pan on a rack for 20 minutes, then invert them onto the rack and turn right side up to cool completely.
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Prepare the frosting, soak, and filling while the cake cools.
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To make the frosting: In a large measuring cup or small bowl, stir together the coconut milk and vanilla; set aside.
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In the bowl of a stand mixer fitted with the flat beater (or in a large mixing bowl using a handheld mixer), cream the butter on medium speed until light and fluffy, about 2 minutes.
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Gradually add 3 cups (339g) of the confectioners’ sugar then the coconut milk mixture. Mix on low speed until smooth and creamy, 1 to 2 minutes.
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Gradually add the remaining 2 1/2 to 3 cups (283g to 339g) confectioners’ sugar and the salt, beating until the frosting is light and fluffy, about 2 minutes.
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The frosting can be stored in an airtight container at room temperature for up to two days or refrigerated for up to five days; bring it to room temperature before using.
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To make the soak: In a measuring cup, stir together the coconut milk and vanilla; set aside.
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To make the filling: In a medium bowl, stir together the coconut and heavy cream. Let stand for about 10 minutes, then stir in 1/2 cup of the prepared frosting. Use immediately; the filling shouldn’t be prepared ahead.
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To assemble the cake: Level the tops of the layers with a serrated knife so they’re flat. Brush the cut side of the layers with the soak.
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Place one layer, cut side up, on a flat serving plate; you can keep the edges of the plate clean by sliding strips of parchment under the cake while you frost it. Spread the filling over the top of the layer.
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Place the second layer, cut side down, on top of the filling and frost the top and sides of the cake with the frosting.
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Sprinkle some of the sweetened shredded coconut on top of the cake and press the rest into the sides, covering the entire cake with coconut.
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Storage information: The cake can be stored, loosely covered, at room temperature or in the refrigerator for up to three days. If refrigerated, bring the cake to room temperature before serving.
