Coconut-Key Lime Sheet Cake

Position rack in middle of oven; preheat to 350 °F. Spread coconut in a single layer on a rimmed baking sheet and toast until golden brown and aromatic, 3 –5 minutes. Let cool 5 minutes. Meanwhile, coat a 13x9" pan with nonstick spray. Dust with flour, tapping out any excess. Whisk toasted coconut, baking powder, salt, and 1 3/4 cups flour in a medium bowl. Mix coconut oil, vanilla, and 2/3 cup coconut milk in another medium bowl. Cream sugar and butter in the bowl of a stand mixer fitted with the paddle attachment at medium speed, occasionally scraping down sides of bowl, until pale and fluffy, 3 –4 minutes. Add eggs, one at a time, beating to blend after each addition. Reduce mixer speed to low and beat in dry ingredients in 3 additions, alternating with coconut milk mixture in 2 additions, beginning and ending with dry ingredients. Pour batter into prepared pan, smoothing top with a spatula. Bake cake until top is set and golden brown and a tester inserted into the center comes out clean, 25 –30 minutes. Mix key lime zest, juice, and remaining 2/3 cup coconut milk in a small bowl (it's okay if mixture separates slightly; it will be absorbed by cake). Using a skewer, poke holes every 2" into warm cake. Slowly pour key lime zest mixture over cake. Transfer pan to a wire rack and let cake cool completely.