Prepare the Oven and Pan: Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan with butter or baking spray, ensuring the surface is coated.
Combine the Dry Ingredients: In a medium mixing bowl, whisk together the all-purpose flour, baking powder, and salt until evenly distributed.
Cream the Butter: Place the softened butter in a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed until creamy and smooth.
Add the Sugar: Add the granulated sugar to the butter. Continue beating for about 3 minutes, until the mixture becomes light and fluffy.
Incorporate the Eggs: Add the eggs one at a time, beating well after each addition. Scrape down the sides of the bowl as needed to ensure the ingredients are evenly combined.
Mix in the Sour Cream and Vanilla: Add the room-temperature sour cream and vanilla extract. Beat until the mixture is smooth and evenly blended.
Add the Dry Mixture: Reduce the mixer speed to low. Gradually add the flour mixture, mixing only until the dry ingredients are incorporated. Avoid overmixing to help maintain a tender cake texture.
Fold in the Peaches: Add approximately three-quarters of the prepared peaches to the batter. Gently fold them in with a rubber spatula until distributed throughout.
Fill the Cake Pan: Transfer the batter to the prepared cake pan. Spread it evenly, making sure the batter reaches the edges of the pan.
Add the Peach Topping: Scatter the remaining peaches over the surface of the batter. If desired, sprinkle a little extra granulated sugar over the top for a lightly crisp, golden finish.
Bake the Cake: Place the pan in the preheated oven and bake for 45 minutes, or until a wooden pick inserted into the center comes out clean.
Cool Before Serving: Remove the cake from the oven and let it cool for at least 30 minutes before slicing and serving.
Store the Cake: Once cooled, cover the cake and keep it at room temperature for up to 3 days.