Preheat oven to 350°F (175°C). Grease and line the bottom of a 9-inch round cake pan with parchment paper.

Make the brown butter: Melt 1/2 cup butter in a small saucepan over medium heat. Continue cooking, swirling occasionally, until foam subsides and brown specks form at the bottom, releasing a nutty aroma (about 5-7 minutes). Remove from heat immediately and pour the browned butter, including the sediment, into a small bowl. Whisk in the brown sugar until combined.

Pour the brown sugar-butter mixture evenly into the bottom of the prepared cake pan. Arrange the thin nectarine slices decoratively over the topping in a circular pattern.

Prepare the cake batter: In a medium bowl, whisk together the flour, baking powder, and salt. In a separate large bowl, cream the 1/2 cup softened butter and granulated sugar until light and fluffy. Beat in the eggs one at a time, then stir in the vanilla extract.

Alternately add the dry ingredients and the buttermilk to the butter mixture, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.

Gently spoon the cake batter evenly over the nectarines and topping. Bake for 45-50 minutes, or until a toothpick inserted into the center comes out clean.

Let the cake cool in the pan on a wire rack for 15 minutes. Place a serving plate upside down over the cake pan. Carefully invert the cake onto the plate so the topping is now on top. Remove the pan and peel off the parchment paper. Cool completely before slicing and serving.