Vanilla bean coconut cake

Reduced coconut milk:
2 13-to 14-ounce cans unsweetened coconut milk* (preferably organic)
Cake:
2 cups all purpose flour
2 1/4 teaspoons baking powder
1/2 teaspoon salt
3/4 cup (1 1/2 sticks) unsalted butter, room temperature

1 cup sugar
3 large eggs
Seeds scraped from 1 split vanilla bean or 1 1/2 teaspoons vanilla extract
1 cup reduced coconut milk (see above), room temperature
Frosting:
1/2 cup (1 stick) unsalted butter, room temperature
1 cup powdered sugar
1/3 cup reduced coconut milk (see above), room temperature
Seeds scraped from 1 split vanilla bean or 1 1/2 teaspoons vanilla extract
1/8 teaspoon salt
1 1/2 cups sweetened flaked coconut, lightly toasted (for garnish)

For reduced coconut milk:
Bring coconut milk to boil in large deep saucepan over medium-high heat (coconut milk will boil up high in pan). Reduce heat to medium-low; boil until reduced to 1 1/2 cups, stirring occasionally, about 45 minutes. Remove from heat; cool completely. Transfer to small bowl. Cover; chill (coconut milk will settle slightly as it cools). DO AHEAD: Can be made 2 days ahead. Keep chilled.

For cake:
Position rack in center of oven; preheat to 350°F. Butter and flour two 8 or 9 in cake pans. Cut out parchment paper circles the size of each pan, butter and flour and place in the bottom of the pans. Whisk flour, baking powder, and salt in medium bowl. Using electric mixer, beat butter in large bowl until smooth. Add sugar; beat on medium-high speed until well blended, about 2 minutes. Add 2 eggs, 1 at a time, beating well after each addition and occasionally scraping down sides of bowl. Beat in seeds from vanilla bean and remaining egg. Add half of flour mixture; mix on low speed just until blended. Add 1 cup reduced coconut milk; mix just until blended. Add remaining flour mixture; mix on low speed just until blended. Divide batter among cake pans. Bake until tops spring back when gently touched and tester inserted into center comes out clean, about 35 minutes. Transfer cakes in pans to rack; cool 10 minutes. Carefully remove cakes from pans, peel off the parchment and cool completely on rack.

For frosting:
Using electric mixer, beat butter in large bowl until smooth. Add sugar, 1/3 cup reduced coconut milk, seeds from vanilla bean, and salt. Beat on medium-low speed until blended, scraping down sides of bowl. Increase to medium-high and beat until light and fluffy. Using small offset spatula or a knife, frost first cake layer. Top with second layer, frost top and side of cake. Sprinkle with coconut.

Recipe adapted from Epicurious