Preheat the oven to 350 degrees. Spray a 9-inch cake pan with nonstick cooking spray, then line the bottom with a parchment paper circle.
In a small bowl, mix the brown sugar with the melted portion of the butter until it forms a thick paste. Spread it evenly over the parchment paper.
Arrange the sliced peaches in a snug single layer over the brown sugar mixture.
In a large mixing bowl, beat the remaining butter with the granulated sugar until light and fluffy, scraping down the sides often, about 6 minutes.
Add the egg and vanilla extract to the butter mixture and mix until smooth.
In a separate small bowl, whisk together the flour, baking powder, and salt.
Add the flour mixture to the batter a scoop at a time, alternating with the milk, and mix just until combined.
Spoon the batter evenly over the peaches and gently smooth it to the edges of the pan.
Bake for 50 to 55 minutes, or until a toothpick inserted in the center comes out clean.
Place the cake pan on a wire rack and let the cake cool for 10 minutes.
Run a knife around the edge of the pan. Place a serving plate over the cake pan and invert the cake in one steady motion.
Carefully lift off the parchment paper and replace any peach slices that may have stuck.
Let the cake cool completely or serve it slightly warm.
