• Preheat oven to 325°F (165°C) with a rack in the middle position. Grease and flour a 9-inch round cake pan.

Peach Cobbler Topping

• Place peaches in a bowl and add light brown sugar, granulated sugar, allspice, and salt. Gently mix together. Set aside.
• In a separate bowl, whisk together flour, baking powder, baking soda, and salt. Sift to remove any lumps. Set aside.
• In the bowl of a stand mixer fitted with the paddle attachment or in a large bowl using a hand mixer cream the butter on medium-high speed until creamy and smooth, about 1 minute.
• Turn the mixer speed to low; add the sugar, vanilla extract to the butter and mix until incorporated. Turn the speed to medium-high and beat until light and fluffy, about 5 minutes. Scrape the bowl often for even incorporation.
• Turn the speed to medium-low and add the eggs and egg yolks one at a time, beating just until combined. Scrape the bowl after each addition.
• Add one-third of the flour mixture to the butter mixture and mix on low speed, just until combined. Add half of the sour cream, mixing until incorporated. Repeat with another one-third flour, scraping the bowl and paddle after each addition. Add rest of sour cream, mixing until incorporated, and finish with final third of flour.
• Scrape the batter into the pan with the peaches, smooth the top with the spatula.
• Bake for 55-70 minutes or until it is no longer jiggly and a tester inserted into the cake comes out clean with just a couple lightly moist crumbs.
• Let the cake cool in the pan for 20 minutes, then flip it over onto a serving plate and allow it to cool.
• While the cake is cooling make the brown butter icing.

Brown Butter Icing

• Place the butter in a skillet that is light in color (this helps you monitor the browning process). Melt the butter over low heat, stirring throughout. Stirring the butter with a rubber spatula through the browning process will help it melt evenly.
• After the butter has melted, raise the temperature to medium heat. As the butter continues to cook, the butter will start to foam, bubble, and splatter.
• After about 5-8 minutes, you'll notice the butter beginning to brown, with lightly browned specks forming at the bottom of the skillet and a nutty fragrance.
  
• Once browned, remove the skillet from heat and let it cool for 5 minutes (as the butter will eventually solidify).
• After that, whisk in the remaining icing ingredients until the mixture is smooth. If you want a thicker texture, add more confectioners' sugar, or add more milk if you want to thin it out. Finally, drizzle the icing over the cake before slicing and serving.