Kokos- och kardemummakaka

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1. 1
   Place a rack in middle of oven and preheat to 325°. Grease an 8"-diameter cake pan with unsalted butter and line with a parchment paper round.
2. 2
   Whisk 1⅓ cups (167 g) all-purpose flour, 1¼ tsp. baking powder, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and ½ tsp. ground cardamom in a medium bowl to combine.
3. 3
   Beat 1 cup (200 g) granulated sugar and ½ cup (1 stick) unsalted butter, room temperature, in the bowl of a stand mixer fitted with the paddle attachment on medium speed until light and fluffy, about 3 minutes. Add 2 large eggs one at a time, beating until incorporated after each addition and scraping down sides of bowl with a rubber spatula as needed. Add 2 Tbsp. vegetable oil, 1 tsp. vanilla extract, and ½ tsp. coconut extract and beat until combined. Reduce speed to low and gradually pour in ½ cup whole milk; beat until incorporated. Scrape down sides of bowl and mix to combine. Add dry ingredients and gently fold in with spatula just until no dry spots remain.
4. 4
   Scrape batter into prepared cake pan; smooth surface. Bake cake until a tester inserted in the center comes out clean, 45–50 minutes. Transfer pan to a wire rack and let cake cool, 1–2 hours. Do ahead: Cake can be baked 2 days ahead. Store tightly wrapped at room temperature.
5. 5
   While the cake is cooling, heat a dry small skillet over medium. Toast 1 cup sweetened shredded coconut, stirring often, until mostly golden brown, 5–7 minutes. Immediately transfer to a small bowl and let cool (coconut will continue to darken slightly as it cools).
6. 6
   Using an electric mixer on medium speed, beat ½ cup (1 stick) unsalted butter, room temperature, in a medium bowl until light and fluffy, about 3 minutes. Add 2 cups (220 g) powdered sugar in 2 additions, beating until combined after each addition. Add ¾ tsp. Diamond Crystal or ½ tsp. Morton kosher salt and ½ tsp. coconut extract and beat until incorporated. Reduce speed to medium-low and beat, scraping down sides of bowl as needed, until mixture is very light and fluffy, about 5 minutes. Add ¼ cup heavy cream and beat until frosting is smooth. Frosting should be spreadable; if it’s too thick, beat in more cream 1 Tbsp. at a time.
7. 7
   Invert cake onto a platter and unmold; carefully remove parchment paper. Using an offset spatula, spread frosting evenly over top and sides of cake. Sprinkle shredded coconut over to coat top and sides; press gently to adhere. Do ahead: Frosting can be made 2 days ahead. Cover and chill. Bring to room temperature, then beat until fluffy again before using.