Preheat oven to 350°F with the rack in the middle. Grease the side and bottom of a 9x3-inch springform pan with butter or cooking spray. Line the bottom with parchment paper, then grease the parchment too.
Sift the flour, baking powder, and baking soda together into a medium bowl.
In a separate bowl, beat the softened butter, sugar, and eggs on high speed for 2 to 3 minutes, until very light in color and fluffy.
Add the vanilla and Greek yogurt, then continue beating for about 1 minute, until creamy and light.
With the mixer on low speed, mix in the flour mixture just until combined. Do not overmix.
Transfer the batter to the prepared springform pan and smooth the top gently.
Arrange the sliced peaches over the batter, then scatter the blueberries evenly in the spaces between the peach slices. Sprinkle the fruit with 1 teaspoon granulated sugar.
Bake for about 1 hour, or until the cake is golden and a tester inserted into the center comes out clean. Halfway through baking, add a few extra peach slices and blueberries on top if desired, then return the cake to the oven.
Let the cake cool in the baking pan on a wire rack. After about 40 minutes, release the cake from the springform pan.
Once cool enough to handle, move the cake with the parchment still attached onto a cake plate. If using a 9-inch round cake pan, cool in the pan for 40 minutes or more, then invert carefully onto a plate.