Step 1: Brown the butter by placing 1/2 cup unsalted butter in a small saucepan over medium heat and melting it completely, stirring occasionally until the milk solids turn golden brown and smell nutty, about 8 to 10 minutes.

Step 2: Pour the brown butter into a small bowl and let it cool to room temperature, about 10 to 15 minutes, to prevent cooking the egg.

Step 3: In a large mixing bowl, combine the cooled brown butter and 3/4 cup granulated sugar, stirring vigorously with a wooden spoon until the mixture is sandy and well combined, about 1 minute.

Step 4: Add 1 large egg and 1 teaspoon vanilla extract to the butter-sugar mixture and stir until smooth and slightly fluffy, about 1 minute.

Step 5: In a separate bowl, whisk together 1 1/2 cups all-purpose flour, 1/2 teaspoon baking soda, and 1/4 teaspoon sea salt until evenly distributed.

Step 6: Pour the dry ingredients into the wet mixture and stir gently with a wooden spoon until just combined and no flour streaks remain.

Step 7: Preheat the oven to 375°F (190°C) and line two baking sheets with parchment paper.

Step 8: Scoop the dough using a cookie scoop or teaspoon onto the prepared baking sheets, spacing cookies about 2 inches apart; the dough balls should be roughly the size of a walnut, yielding about 24 cookies.

Step 9: Bake for 11 to 13 minutes, until the edges are golden brown and centers still look slightly underbaked, then remove from the oven.

Step 10: Let the cookies cool on the baking sheet for 3 to 5 minutes before transferring them to a cooling rack to finish cooling.