Melt 10 tablespoons butter in saucier pan or 10-inch skillet over medium-high heat. Continue to cook, swirling pan constantly, until butter is dark golden brown and has a nutty aroma, 1 - 3 minutes. Transfer browned butter to large heatproof bowl. Add remaining 4 tablespoons butter, in 8 pieces, (carefully as it might splatter) and stir until completely melted; set aside for 15 minutes.

Meanwhile, set shelf to the middle and heat oven to 350 degrees. Line 2 baking sheets with parchment paper. In a shallow dish, mix 1/4 cup brown sugar and granulated sugar 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 and salt to bowl with cooled butter; stir with a wooden spoon until no sugar lumps remain, about 30 seconds. Scrape down bowl; add egg and 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.

Use a medium cookie scoop (1 1/2 tablespoons dough) and roll into balls; place on prepared sheets, spaced 2 inches apart (14 per sheet). Remove each ball and roll in sugar mixture to coat; return to its place on sheet.

Bake cookies, 1 sheet at a time, until browned and edges have begun to set but centers are still soft (cookies will look raw between cracks and seem underdone), 12 - 14 minutes, rotating sheet after 8 minutes. If you're not sure, err on the side of underdone, or they will turn out crispy, rather than chewy. Let cookies cool on sheets for 5 minutes, then transfer parchment to wire rack to cool completely.