Preheat oven to 340 degrees F (170 degrees C). Place butter in a 9-inch round cake pan and melt in the oven as it preheats.
Remove pan from oven and sprinkle brown sugar evenly over melted butter. Arrange nectarine pieces on the brown sugar layer.
Gently whisk flour, baking powder, and salt together in a bowl. Beat unsalted butter and white sugar together in a separate bowl until creamy and smooth; add eggs, 1 at a time, beating well after each addition. Beat vanilla extract into the creamed butter mixture. Add flour mixture to creamed butter mixture; beat on low speed just until batter is combined, about 1 minute. Spread batter over nectarine layer.
Increase oven temperature to 350 degrees F (180 degrees C).
Bake in the preheated oven until cake is golden brown and a toothpick inserted in the center comes out clean, 35 to 40 minutes. Cool cake in pan on a wire rack for 15 minutes. Run a knife around edge of pan and invert cake onto a serving plate. Serve warm or at room temperature.