Set a rack in the middle of the oven and preheat it to 375 degrees.
Melt the 6 tablespoons butter in an ovenproof 9-inch skillet over moderate heat.
Add brown sugar and cook, stirring until it is dissolved and bubbling. Remove from the heat.
Sprinkle macadamia nut pieces over the bottom of the skillet or cake pan.
Arrange papaya wedges in a tight pinwheel around the outside of the pan, filling in the center with any broken or odd-shaped pieces.
Sift flour, baking powder, baking soda, and salt onto a sheet of wax paper.
Beat the remaining 8 tablespoons of butter and the sugar in an electric mixer at high speed for 2 minutes, or until well combined and smooth.
Add eggs one at a time, beating until each is incorporated. Continue beating 5 more minutes. With the mixer on the lowest setting, or using a rubber spatula, beat or fold in one-third of the flour mixture.
Beat or fold in vanilla extract and half of the buttermilk, then another one-third of the flour mixture.
Beat or fold in the remaining buttermilk and then the remaining flour mixture.
Spread batter evenly over the papaya wedges.
Bake for 30 minutes or until golden and the center springs back when lightly pressed.
Cool the cake in the pan on a wire rack for 5 minutes.
Run a knife around the edge of the pan and place a serving platter on top of the pan. Carefully flip the entire thing over. If any papaya is stuck to the pan, scrape it off with a knife and rearrange on top of the cake.
Let cool for 15 minutes before serving.