Drain pineapple rings and discard juice. Place pineapple on a paper-towel-lined plate to drain, turning once, for 5 minutes.
Melt 6 tablespoons butter and transfer to a 9-inch round cake pan to evenly coat bottom; sprinkle brown sugar evenly over bottom of cake pan and top with an even layer of pineapple rings and pecans, if using. Set prepared cake pan aside.
In a large bowl, whisk together flour, baking powder, and salt; set aside. In a small bowl, whisk together eggs, egg yolks, and vanilla until well combined; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat remaining 6 tablespoons butter with granulated sugar on medium-high speed until light and fluffy. Reduce speed to low and add flour mixture alternating with egg mixture; beat until batter is just combined.
Pour batter in prepared cake pan and transfer to oven. Cook until a cake tester inserted into the center comes out clean, about 1 hour and 10 minutes. Transfer to a wire rack to cool for 10 minutes.
Run a small sharp knife around edge of pan to release cake. Place a plate on top of cake pan and invert cake onto plate; serve.