Preheat your oven to 350°F (175°C) and prepare a 9-inch springform pan by greasing it with butter and lining the base with parchment paper.
In a bowl, sift together the flour, baking powder, and baking soda. Set this dry mixture aside for later use.
In a large mixing bowl, beat the softened butter and sugar together until the mixture becomes light and fluffy, about 3-4 minutes. Then add in the eggs one at a time, mixing well after each addition. Incorporate the vanilla extract and Greek yogurt, mixing until smooth.
Gently fold the dry ingredients into the wet mixture until just combined; be careful not to overmix.
Pour the batter into the prepared springform pan, then elegantly arrange the peach slices on top, followed by evenly distributing the blueberries. Finally, sprinkle the brown sugar over the fruit layer.
Bake in the preheated oven for about 50-60 minutes or until the cake is golden brown and a toothpick inserted in the center comes out clean. Rotate the pan halfway through baking to ensure even cooking.
Allow the cake to cool in the pan for 30 minutes before carefully releasing it from the springform. Transfer the cake to a wire rack to cool completely before slicing.