To make the coconut cake layers:

1. Using an electric mixer (a stand mixer or handheld), beat the butter and sugar at medium speed until light and creamy, about 3 minutes.
2. Add eggs two at a time, mixing on low speed to incorporate after each addition.
3. Add coconut milk and vanilla and mix at medium speed.
4. Add flour and baking powder and mix on the lowest speed until just combined.
5. Divide batter between three greased 6" cake pans and bake at 325F or 160C for 30 minutes or until cakes spring back up when poked in the middle.
6. Leave cakes in their pans for 10 minutes before turning out onto a cooling rack and let them cool completely.

To make the coconut milk filling and frosting:

1. Put flour, milk, coconut milk and salt in a pan and cook over medium heat for about 5 minutes. Whisk it constantly until it simmers to prevent lumps. When the mixture thickens, switch to a spatula. Scrape along the bottom and sides of the pan, stirring for about two minutes until the mixture thickens enough to leave ribbons when you lift the spatula.
2. Remove the pan from the heat and stir in vanilla. Pour the mixture into a bowl and cover it with plastic wrap (e.g. cling film / Saran Wrap), pressing it down to touch the surface of the mixture. Put it into the fridge for about 30 minutes to cool to room temperature. 
3. Mix butter and sugar at medium speed until creamy and lighter in colour, about 3 minutes. Add cooled flour mixture and mix for another minute, until smooth.