Preheat oven to 325F. Grease and line the bottom of an 8" pan with parchment paper, set aside.
Using a stand mixer with paddle attachment, cream together the butter, coconut oil, and sugar and until smooth. Stir in the papaya puree and milk until smooth. Whisk in the dry ingredients until just combined, be careful not to overmix!
In a medium size bowl, with a hand mixer whip the egg whites and cream of tartar until medium stiff peaks form. Gently fold the egg whites into the cake batter.
Divide the batter into pans. Bake for about 18-20 minutes or until a toothpick comes out clean. Allow to come to room temperature.
Make the papaya layer: Heat papaya puree, lime juice, sugar, and cornstarch in a saucepan over medium heat until thickened (5–8 mins) be careful not to overcook. Cool completely.
For the frosting: using a stand or handheld mixer, beat butter, cream cheese, vanilla, and powdered sugar until smooth.
Assemble: Evenly spread the cooled papaya puree layer over the cooled cake. Top off with the frosting and decorate with a little bit of the papaya puree. Sprinkle with desiccated coconut and enjoy!