Preheat the oven to 350°F (175°C). Grease and flour a 9x13-inch baking pan.

Mix Dry Ingredients: In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.

Brown the Butter: In a saucepan over medium heat, melt the 1 cup butter. Continue cooking, swirling occasionally, until the butter turns a golden brown color and has a nutty aroma (about 5-7 minutes). Remove from heat and let cool slightly.

Combine Butter and Sugar: In a large mixing bowl, combine the brown butter and granulated sugar. Beat with an electric mixer on medium speed until well combined.

Add Eggs and Vanilla: Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.

Finish Batter: Gradually add the dry ingredients to the butter mixture, alternating with the buttermilk. Start and end with the flour mixture. Mix until just combined.

Bake Cake: Pour the batter into the prepared baking pan and smooth the top. Bake for 30-35 minutes, or until a toothpick inserted in the center comes out clean.

Cool: Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.