Preheat the oven to 350°F (175°C) and grease a 9-inch round cake pan with butter or non-stick spray.

In a medium mixing bowl, whisk together the all-purpose flour, baking powder, and salt until aerated and combined.

In a separate large bowl, cream the softened butter and 0.875 cups of the granulated sugar together using an electric mixer on medium-high speed until light and fluffy.

Incorporate the egg and vanilla extract into the butter mixture, beating until fully emulsified.

Reduce mixer speed to low and add the dry ingredients in three parts, alternating with the milk, beginning and ending with the flour mixture.

Transfer the batter into the prepared cake pan and use an offset spatula to level the surface.

Arrange the peach wedges on top of the batter in a concentric circular pattern, pressing them slightly into the surface.

Sprinkle the remaining 0.125 cups of granulated sugar evenly over the arranged peach slices.

Bake for 45 to 50 minutes, or until the cake is golden brown and a toothpick inserted into the center comes out clean.

Remove from the oven and allow the cake to cool in the pan for at least 15 minutes on a wire rack before serving.