1. Heat 10 tablespoons butter in 10-inch skillet over medium-high heat until melted, about 2 minutes. Continue to cook, swirling pan constantly until butter is dark golden brown and has nutty aroma, 1 to 3 minutes. Remove skillet from heat and transfer browned butter to large heatproof bowl. Stir remaining 4 tablespoons butter into hot butter to melt; set aside for 15 minutes.

2. Meanwhile, adjust oven rack to middle position and heat oven to 350 degrees. Line 2 large (18 by 12-inch) baking sheets with parchment paper. In shallow baking dish or pie plate, mix granulated sugar and 1/4 cup packed brown sugar, rubbing between fingers, until well combined; set aside. Whisk flour, baking soda, and baking powder together in medium bowl; set aside.

3. Add remaining 1 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 with rubber spatula; 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 with rubber spatula to ensure that no flour pockets remain and ingredients are evenly distributed.

4. Divide dough into 24 portions, each about 2 tablespoons, rolling between hands into balls about 1 1/2 inches in diameter. Working in batches, toss balls in reserved sugar mixture to coat and set on prepared baking sheet, spacing them about 2 inches apart, 12 dough balls per sheet. (Smaller baking sheets can be used, but it will take 3 batches.)

5. 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; see photo below), 12 to 14 minutes, rotating baking sheet halfway through baking. Do not overbake.

6. Cool cookies on baking sheet 5 minutes; using wide metal spatula, transfer cookies to wire rack and cool to room temperature.