Preheat oven to 350 °F and grease and line an 8-inch square pan with parchment paper.
Spread the coconut onto a baking sheet and toast until outside edges of pan have browned. Remove from the oven. Mix the coconut around on the baking sheet to mix in the browned coconut and set it aside to cool.
Increase oven temperature to 375 °F.
In a bowl, whisk vegetable oil, 1 cup of coconut milk and egg yolks until evenly blended.
In a separate bowl, sift flour, 1 cup of sugar, baking powder and salt.
Add flour to coconut milk mixture and stir just until blended. Stir in 1 1/3 cups toasted coconut. Reserve the rest of the coconut for topping the cake (after it has baked).
In a third bowl, whip egg whites with remaining ½ cup sugar until they hold a soft peak. Fold a third of the whites into the cake mix (dry ingredients) until incorporated then fold in remaining whipped whites.
Spoon batter into prepared pan and bake for 25 to 30 minutes, until top turns light brown and springs back when touched.
While still warm, brush top of cake with remaining 3 Tbsp coconut milk and sprinkle with remaining 1 cup of toasted coconut. Let cool completely before slicing.