Brown the Butter: Melt the unsalted butter in a saucepan over medium-low heat, then raise the heat to medium and cook while stirring constantly. The butter will sizzle and pop as it cooks. When the popping slows or stops and the butter begins turning golden brown, with brown flecks visible on the bottom of the pan, immediately remove it from the heat. Cool the Brown Butter: Pour the browned butter into a large heatproof mixing bowl and let it cool completely until it is no longer warm to the touch before continuing.
Prepare the Oven and Pan: Preheat the oven to 375°F (190°C) and line a 12-count muffin tin with paper liners.
Combine the Wet Ingredients: Once the browned butter has cooled, add the granulated sugar, egg, and vanilla extract. Whisk thoroughly until the mixture is well combined.
Add the Buttermilk: Pour the buttermilk into the butter mixture and stir until incorporated.
Mix the Dry Ingredients: In a separate bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until evenly combined.
Begin Combining the Batter: Add the dry ingredients to the wet mixture and gently fold with a spatula until the flour is only about halfway incorporated.
Fold in the Peaches: Add the diced peaches and continue folding just until the ingredients are completely combined. Avoid over-mixing, as this can make the finished muffins dense.
Fill the Muffin Tin: Divide the batter evenly among the prepared muffin liners, then sprinkle the tops with coarse sugar or regular granulated sugar.
Bake the Muffins: Place the muffin tin on the center rack of the 375°F (190°C) oven and bake for 20 minutes, or until a toothpick inserted into the center comes out clean or with moist crumbs.
Cool in the Pan: Leave the baked muffins in the muffin tin for 5 minutes before carefully transferring them to a cooling rack.
Finish with Glaze: Allow the muffins to cool completely before drizzling them with vanilla glaze, if desired.
