Heat a nonstick skillet over medium high heat. Add 10 tablespoons butter and melt. Continue to cook, stirring constantly with a spatula till butter browns, about 4 minutes. Watch carefully so butter does not burn.
Remove skillet from heat and transfer browned butter to medium mixing bowl. Add remaining 4 tablespoons butter and stir to melt; let sit for 15 minutes to cool.
Preheat oven to 350°F and place rack in the middle. Line two large baking sheets with parchment paper.
In a shallow dish, combine granulated sugar and 1/4 cup packed brown sugar; set aside.
Whisk together flour, baking soda, and baking powder; set aside.
Add remaining 1-3/4 cups packed brown sugar and salt to bowl with cooled butter; mix well (till no lumps remain). Add egg, egg yolk, and vanilla and mix till fully incorporated.
Add flour mixture and mix till just combined, about a minute.
Form dough into balls, about 3/4 inch in diameter (about 1 tablespoon each). Roll balls in reserved sugar mixture, coating well. Place on prepared baking sheets about 2 inches apart (they will spread a bit when baked).
Bake one sheet at a time, till cookies are browned and still puffy and edges have just begun to set, about 12 minutes. Cookies will look underbaked. Do not overbake.
Remove cookies from oven and cool on baking sheet for 5 minutes, then transfer to wire rack and cool to room temperature.