Preheat oven to 325 degrees. Place 4 tablespoons butter in a 9-inch round cake pan (2 inches deep) and melt in oven. Remove pan from oven and sprinkle brown sugar evenly over butter.
With two double-thick layers of paper towels, press as much liquid as possible out of nectarines. Arrange fruit in an even layer in pan.
In a medium bowl, stir together flour, baking powder, baking soda, and salt. In a large bowl, using an electric mixer, beat 6 tablespoons butter on high until light and fluffy. Add granulated sugar and beat until well combined. Beat in eggs, one at a time, scraping down bowl as needed. Beat in vanilla. With mixer on low, add flour mixture in three additions, alternating with two additions yogurt, and beat to combine.
With a spatula, spread batter over nectarines. Bake until cake is dark golden brown and a toothpick inserted in center comes out clean, 1 hour 10 minutes. Let cake cool in pan on a wire rack, 15 minutes. Run a knife around edge of pan and invert cake onto a serving plate. Serve warm or at room temperature.
Pressing excess liquid from the thawed fruit makes all the difference in the texture of this cake, so don't rush that step.