Start by melting 10 tablespoons of butter in a medium saucepan over medium heat. Stir frequently until the butter turns golden brown and releases a nutty aroma. This process usually takes about 5-7 minutes. Once browned, remove it from the heat and let it cool slightly.
In a large mixing bowl, whisk together 2 cups of all-purpose flour, 1 teaspoon of baking soda, 1 teaspoon of baking powder, and 1 teaspoon of salt.
In a separate bowl, combine the browned butter and 1.75 cups of brown sugar. Mix until well combined.
Add 1 large egg, 1 egg yolk, and 1 teaspoon of vanilla extract to the butter-sugar mixture. Beat until creamy.
Gradually add the dry ingredients to the wet mixture, folding in the flour until no dry streaks remain.
Cover the bowl with plastic wrap and refrigerate the dough for at least 30 minutes.
Preheat your oven to 350°F (175°C) and line your baking sheet with parchment paper.
Scoop the dough using a cookie scoop or tablespoon. Roll in a mixture of 1 cup of brown sugar and 1 cup of granulated sugar.
Place the dough balls on the prepared baking sheet, spacing them about 2 inches apart. Bake for 10-12 minutes, until edges are set and centers are slightly soft.
Remove from the oven and immediately sprinkle with sea salt. Cool on the baking sheet before transferring to a wire rack.