Preheat the oven to 350ºF. Lightly spray the bottom of a 9 x 13-inch baking pan with non-stick baking spray; set aside.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt; set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream the butter and sugar at medium speed until light and fluffy, about 3 minutes. Scrape down the sides of the bowl and add the egg yolks, one at a time, beating well after each addition. Add the extracts, beating until combined. Gradually add the flour mixture to the butter mixture, alternating with the coconut milk, beginning and ending with the flour mixture. Scrape down the sides of the bowl and beat until combined.
In a medium bowl using a hand-held mixer, beat the egg whites at high speed until stiff peaks form, being careful not to over beat. Fold the egg whites into the batter. Gently spread the batter into the prepared baking pan.
Bake until a toothpick inserted in the center of the cake comes out clean, about 25-30 minutes. Allow the cake to cool completely in the pan on a wire rack.