Preheat oven to 350°F (175°C). Grease a 9-inch round cake pan.
In a small saucepan, melt butter and brown sugar until smooth and bubbling. Pour into the pan, tilting to coat evenly. Arrange peach slices in a fan or loose pattern over the caramel.
In a medium bowl, whisk flour, baking powder, baking soda, and salt.
In a large bowl, beat softened butter and sugar until light and fluffy. Beat in eggs one at a time, then stir in vanilla.
Mix in sour cream, then alternate adding dry ingredients and milk until just combined. Batter should be thick but spreadable.
Gently spread batter over the peaches, smoothing the top.
Bake 40–45 minutes, until golden and a tester comes out clean.
Cool in pan for 10 minutes, then run a knife around the edge and invert onto a plate. Peel away any stuck peaches and patch them back on — imperfect is beautiful.