Heat 10 tablespoons butter in a 10" 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, 1-3 minutes.

Remove skillet from heat and transfer browned butter to large bowl.

Stir remaining 4 tablespoons of butter into hot butter to melt; set aside for 15 minutes.

Heat oven to 350 degrees. Line 2 large 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.

Add remaining 1 3/4 cups brown sugar to bowl with cooled butter; mix until no sugar lumps remain, about 30 seconds.

Scrape down the 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 a final stir with rubber spatula to ensure that no flour pockets remain and ingredients are evenly distributed.

Divide dough into 24 portions, each about 2 tablespoons, rolling between hands into ball about 1 1/2" in diameter.

Working in batches, toss balls in reserved sugar mixture to coat, and set on prepared baking sheet, spacing them about 2" apart, 12 dough balls per sheet.

(Smaller baking sheets can be used, but it will take 3 batches.)

Bake one sheet at a time until cookies are browned and still puffy, and edges have begun to set but centers are still soft, 12 to 14 minutes, rotating baking sheet halfway through baking.

(Cookies will look raw between cracks and seem underdone; do not overbake.)

Remove cookies from oven and sprinkle salt onto cookies.

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