Use a 9x3-inch springform pan.
Preheat oven to 350°F with rack in middle. Grease the side and the bottom of the pan with butter or cooking spray. Line the bottom of the pan with parchment paper and grease the parchment paper too.
Sift flour, baking powder, and baking soda together into a medium bowl.
In a separate bowl, beat butter, sugar, and eggs until very light in color and fluffy, about 2-3 minutes on high speed. Add vanilla and Greek yogurt and continue beating until very creamy and light, for about 1 more minute.
Keeping the mixer speed low, mix in the flour mixture until combined. Do not overmix.
Transfer the cake batter to the springform pan. Top with sliced peaches, and scatter blueberries evenly on top in the spaces between the peach slices. Sprinkle the fruit with 1 teaspoon granulated sugar.
Bake until the cake turns golden and a tester comes out clean in the center, about 1 hour, depending on your oven. Midway through baking, add some extra peach slices and blueberries on top for prettiness.
When the cake is done baking, let it cool (still in the baking pan) on a wire rack. After about 40 minutes, release the cake from the springform pan.
If using a 9-inch round cake pan, cook in the pan for about 40 minutes or more (up to 1 hour, depending on your oven), then invert onto a plate.