Melt 10 tablespoons butter in 10-inch skillet over medium-high heat. Do not use a nonstick/dark pan or it will be difficult to see the color change. Continue cooking, swirling pan constantly, until butter is dark golden brown and has nutty aroma, 1 to 3 minutes. Transfer browned butter to large heatproof bowl. Add remaining 4 tablespoons butter and stir until completely melted. Set aside for 15 minutes.

Meanwhile, adjust oven rack to middle position and heat oven to 350 degrees. Line 2 baking sheets with parchment paper. In shallow baking dish or pie plate, mix the 2 Tablespoons of dark brown sugar and 2 Tablespoons of granulated sugar until well combined; set aside. Whisk flour, baking soda, and baking powder together in medium bowl; set aside.

Add the 1 and 3/4 cups brown sugar and salt to bowl with cooled butter; mix until no sugar lumps remain, about 30 seconds. Scrape down sides of bowl; add egg, yolk, and vanilla and mix until fully incorporated, about 30 seconds. Scrape down bowl. Add flour mixture and mix until just combined, about 1 minute. Give dough final stir to ensure that no flour pockets remain.

Working with 2 Tablespoons at a time, roll into balls. Roll half of the dough balls into the sugar mixture to coat (you can make more if you run out). Place dough balls 2 inches apart on the baking sheet, repeat. I was able to fit 8 balls on a sheet.

Bake one sheet at a time until cookies are browned and still puffy and edges have begun to set but centers are still soft (cookies will look raw between cracks and seem underdone), 12 to 14 minutes (mine baked for 12 minutes total), rotating baking sheet halfway through baking. Do not overbake. Let cookies cool on baking sheet for 5 minutes. Transfer cookies to wire rack and cool to room temperature.