Preheat the oven to 350ºF (180ºC). Line the bottom of an 8-inch round cake pan with parchment paper. Grease and flour the insides.
In a medium-sized bowl, whisk together the flour, baking powder and salt until well blended.
In a large bowl, beat the butter and sugar together, with an electric mixer on medium speed, until pale and fluffy. Gradually add the oil and coconut extract and beat until smooth and creamy. Beat in the egg whites in 3 separate additions, making sure each one is incorporated before adding the next.
On low speed or by hand, stir in the flour mixture in 2 additions, followed by half of the coconut milk after each addition. Mix just until combined.
Transfer the batter to the prepared cake pan. Bake until a toothpick inserted into the center of the cake comes out clean, about 23-28 minutes, being careful not to over bake. Let the cake cool in the pan for 10 minutes, then remove it from the pan and transfer it to a cooling rack. Brush the warm cake generously with coconut syrup.