10 tablespoons unsalted butter, room temperature
1/2 cup dark-brown sugar
4 cups frozen peaches, thawed and cut into bite-size pieces
1 and 1/2 cups all-purpose flour
1 and 1/2 teaspoons baking powder
1/2 teaspoon baking soda
1/2 teaspoon salt
3/4 cup granulated sugar
2 large eggs
1 teaspoon pure vanilla extract
1/2 cup plain Greek yogurt
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 peaches. 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 peaches. Bake until cake is dark golden brown and a toothpick inserted in center comes out clean, 1 hour and 10 minutes.
Let cake cool in pan on wire rack, 15 minutes. Run a knife around edge of pan and invert cake onto a serving platter.
Serve warm or at room temperature.