Preheat your oven to 350°F (175°C). Grease and flour a bundt pan.
In a bowl, whisk together flour, baking powder, baking soda, and salt.
In another bowl, beat butter and sugar until light and fluffy. Add eggs one at a time, beating well after each addition. Mix in vanilla extract.
Gradually add the dry ingredients to the wet ingredients, alternating with sour cream. Fold in the diced peaches.
Pour the batter into the prepared bundt pan. Bake for 60 minutes or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan for 15 minutes, then invert onto a wire rack to cool completely.
For the icing, melt the butter in a saucepan over medium heat until it turns golden brown. Remove from heat and let cool slightly.
In a bowl, whisk together the browned butter, powdered sugar, milk, and vanilla extract until smooth. Drizzle over the cooled cake.