Prepare the Cake Batter:

Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking pan. In a medium bowl, whisk together flour, baking powder, baking soda, and salt. In a separate large bowl, cream together the butter and sugar until light and fluffy. Add the eggs, one at a time, beating well after each addition. Mix in the vanilla extract and sour cream. Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Fold in the peaches.

Bake the Cake:

Pour the batter into the prepared baking pan and spread evenly. Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool completely.

Make the Brown Sugar Frosting:

In a saucepan over medium heat, melt the butter. Stir in the brown sugar and cook for 2-3 minutes, until it dissolves and becomes smooth. Add the milk, bring to a boil, and then remove from heat. Let the mixture cool slightly before whisking in the powdered sugar and vanilla extract until smooth and creamy.

Frost the Cake:

Spread the frosting evenly over the cooled cake using a spatula. Let the frosting set before slicing.