Creating your Brown Butter Peach Cupcakes is straightforward if you follow these steps:

1. Preheat the Oven: Preheat your oven to 350°F (175°C). Line a muffin tin with cupcake liners.
2. Brown the Butter: In a saucepan over medium heat, melt the unsalted butter. Continue cooking until it turns golden brown and develops a nutty aroma. Remove from heat and let it cool slightly.
3. Mix Dry Ingredients: In a large bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt.
4. Combine Wet Ingredients: In another bowl, beat the brown butter, granulated sugar, and eggs until well combined. Stir in the vanilla extract and milk.
5. Combine Mixtures: Gradually add the dry ingredients to the wet mixture, stirring until just combined. Do not overmix.
6. Fold in Peaches: Gently fold in the diced peaches until evenly distributed throughout the batter.
7. Fill Cupcake Liners: Divide the batter evenly among the cupcake liners, filling each about two-thirds full.
8. Bake: Place the muffin tin in the preheated oven and bake for 20-25 minutes or until a toothpick inserted in the center comes out clean.
9. Cool the Cupcakes: Once baked, remove the cupcakes from the oven and allow them to cool in the tin for 10 minutes. Then, transfer them to a wire rack to cool completely.
10. Prepare the Frosting: In a large bowl, beat the softened butter until creamy. Gradually add the powdered sugar, heavy cream, and vanilla extract. Mix until smooth and fluffy. Finally, fold in the peach puree until well incorporated.
11. Frost the Cupcakes: Once the cupcakes are completely cool, generously frost each cupcake with the peach buttercream frosting using a piping bag or a spatula.