How to Make It
Preheat the oven to 350°. Butter an 8-inch round cake pan. Sprinkle the bottom with 2 tablespoons of the brown sugar.
In a large skillet, combine the remaining 3/4 cup of brown sugar with the 1/2 stick of butter and the vanilla bean and seeds and cook over moderately low heat until the butter is melted. Add the pineapple and cook over moderately low heat, stirring occasionally, until the pineapple is tender, about 20 minutes. Using a slotted spoon, arrange the slices in the cake pan, overlapping them if necessary. Remove the vanilla bean and pour the pan juices over the pineapple.
In a bowl, whisk 1/4 cup of the sour cream with the eggs and vanilla. In another bowl, beat the flour, granulated sugar, baking powder, baking soda and salt. Add the remaining stick of butter and 1/4 cup of sour cream and beat at low speed until smooth, then beat at medium speed until fluffy. Add the sour cream–egg mixture and beat again until fluffy, 2 minutes.
Spoon the batter over the pineapple and spread it evenly. Bake for about 40 minutes, until the cake is deep golden. Let cool for 5 minutes on a rack. Run a knife around the edge of the cake, invert it onto a plate and remove the pan. Replace any pineapple that may have stuck to the pan. Serve warm or at room temperature.