Ombré Coconut Burfi Cake

Preheat oven to 350°F. Line 3 (6-inch) round cake pans with parchment paper; grease pans with butter, and dust with flour. Set aside. Whisk together flour, baking powder, 1 1/2 teaspoons cardamom, and 1 teaspoon salt in a medium bowl. Combine 1 3/4 cups sugar and 3/4 cup butter in a stand mixer fitted with the paddle attachment; beat on medium speed until light and fluffy, 6 to 8 minutes, stopping to scrape sides of bowl as needed. Add eggs, 1 at a time, beating well after each addition. Add egg yolk, and beat until mixture lightens and turns pale yellow, about 3 minutes.

Add flour mixture to sugar mixture in 3 additions alternately with coconut milk, beginning and ending with flour mixture, beating on low speed until combined after each addition. Stir in 1 1/2 teaspoons vanilla. Pour batter evenly into prepared cake pans. Gently tap each pan on counter to release any air bubbles. Bake in preheated oven until a wooden pick inserted in centers comes out clean, 38 to 42 minutes. Let cakes cool in pans 10 minutes. Invert cakes onto a wire rack, and let cool completely, about 1 hour.

Meanwhile, stir together unsweetened shredded coconut, sweetened condensed milk, 1/4 teaspoon salt, and remaining 3/4 teaspoon cardamom in a medium bowl. Set aside.

Combine egg whites, cream of tartar, remaining 3/4 cup sugar, and remaining 1/4 teaspoon salt in bowl of stand mixer. Fill a medium saucepan with water to a depth of 11/2 inches; bring to a simmer over medium. Place bowl with egg white mixture on top of saucepan over simmering water (bottom of bowl should not touch water). Cook egg white mixture, whisking constantly, until sugar dissolves and mixture registers 160°F on an instant-read thermometer, 2 to 4 minutes.

Remove from heat; attach bowl to stand mixer fitted with the whisk attachment. Beat on medium-low speed, gradually increasing speed to medium-high, until mixer bowl no longer feels warm to the touch (about 90°F), 5 to 7 minutes. With mixer on medium-high speed, add remaining 1 1/4 cups butter to mixture, 1 tablespoon at a time, beating well after each addition until frosting is light and fluffy, 8 to 10 minutes. (If mixture appears broken midway through process, continue beating and adding butter to bring it back together.) Beat in remaining 1 teaspoon vanilla. Transfer about 3/4 cup frosting to a piping bag or a ziplock plastic bag with a 1-inch hole snipped in the corner (keep remaining frosting in mixer bowl).

Gently brush off any loose crumbs around sides of cooled cake layers. Place 1 cake layer on a serving plate; pipe a 1-inch border of frosting around top edge of cake layer. Spread half (about 2/3 cup) of the coconut filling in an even layer inside the frosting border. Spread a thin layer of frosting (about 1/8 inch) on top of filling. Top with second cake layer; repeat frosting and filling process. Top with third cake layer. Spread a thin layer (about 3/4 cup) of frosting on top and sides of cake. Chill, uncovered, until frosting is hardened, at least 15 minutes or up to 1 hour.

Meanwhile, stir together 3/4 cup sweetened shredded coconut and 1 small drop of food coloring gel in a small bowl, creating a light purple color. Stir together 3/4 cup sweetened shredded coconut and 2 small drops of food coloring gel in a separate small bowl, creating a dark purple color. Keep remaining 1 cup sweetened shredded coconut undyed.

Spread remaining frosting from mixer bowl on top and sides of chilled cake. Press dark purple–colored coconut around bottom third of cake. Press light purple–colored coconut around middle third of cake. Press undyed coconut around upper third and on top of cake. (You are aiming for an ombré effect.)Serve immediately, or cover and chill up to 1 day; let come to room temperature before serving.