Preheat oven to 350°F. Grease a 9" round cake pan. To make the topping, pour melted butter into pan and sprinkle brown sugar on top, swirling to cover the entire bottom of the pan. Place a single layer of peach slices and blackberries onto the melted butter and sugar mixture and set aside.
In a small bowl, whisk the flour, baking powder, baking soda, and salt. Set aside.
Using a hand mixer or a stand mixer fitted with the paddle attachment, cream the butter with the sugar. Add the Greek yogurt, milk, and vanilla extract and mix until combined.
Add the dry ingredients into the wet ingredients and stir on low speed until just combined.
Pour the batter into the prepared pan, spreading evenly. Place the cake pan onto a baking sheet (optional, but recommended). Bake for 35-40 minutes or until a toothpick inserted into the center of the cake comes out clean.
Allow the cake to cool in the pan for 10 minutes. Use a knife to separate the sides of the cake from the pan, then invert cake onto a plate.