Set a rack in the middle of the oven and preheat it to 375 degrees F.
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.
If you are baking in a 9 inch cake pan instead of the skillet, pour the mixture into the pan and swirl it around to coat the bottom evenly.
Sprinkle the pecan pieces over the bottom of the skillet or cake pan or skillet.
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.
Sift 2 more times to mix and aerate.
Put remaining 8 tablespoons butter and the sugar in the bowl of an electric mixer and beat 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.