Center a rack in the oven and preheat to 350 (F). Line two sheet pans with parchment paper and set aside.
Place butter in a light-colored, heavy-bottomed saucepan over medium-low heat. Melt butter, swirling the pan occasionally, until it starts to foam. Use a heatproof rubber spatula to scrape the sides and bottom of the pan occasionally to prevent the milk solids in the butter from burning. Cook for 5 minutes, or until butter is amber with dark flecks at the bottom of the pan and smells nutty. Remove the saucepan from heat and immediately pour browned butter into a liquid measuring cup. Cool on a wire rack while you work with the other ingredients.
In a medium bowl, whisk together 1 2/3 cups all-purpose flour, 1/2 teaspoon baking powder, 1/2 teaspoon baking soda, and 1 teaspoon kosher salt. In the bowl of a stand mixer fitted with a paddle attachment, combine "tightly packed" 3/4 cup dark brown sugar and 1/4 cup granulated sugar.
Pour the slightly cooled brown butter over the sugars and beat on low. Add 1 large egg and 1 tablespoon pure vanilla extract, continuing to beat on low until just combined. Scrape down the bottom and sides of the bowl with a rubber spatula and gradually add dry ingredients, continuing to beat on low until just combined.
Use a 3-tablespoon cookie dough scoop to portion out cookie dough balls and place them on the prepared sheet pans at least 3 inches apart. Bake for 10 minutes, or until the edges have set but the centers are still gooey. The cookies will look puffed when you pull them out of the oven, but will fall and crack as they cool. Cool cookies on their sheet pan on a wire rack for 20 minutes, or until the edges and bottom of the cookies have set and feel firm to the touch. Serve the cookies warm, or at room temperature. The cookies can be kept at room temperature, in an airtight container or Ziploc bag, for up to 3 days.