Preheat your oven to 350°F (175°C). Prepare a baking sheet with parchment paper.
In a small pot, melt 10 tablespoons of butter over medium heat, stirring constantly. Continue to stir until the butter starts to brown, about 3-5 minutes. Once browned, remove the pot from heat and add the remaining 4 tablespoons of butter. Stir until the additional butter has melted.
Stir in 1.5 cups of brown sugar until smooth. Whisk in the egg, egg yolk, salt, and vanilla extract. Set this mixture aside.
In a separate large bowl, whisk together 1/4 cup of sugar, 1/2 cup of brown sugar, flour, baking soda, and baking powder.
Combine the butter mixture with the dry ingredients, stirring until a dough forms.
Roll the dough into balls. Place 1/4 cup of sugar in a small bowl, and roll each cookie ball in the sugar until coated. Place the sugar-coated dough balls on the prepared baking sheet.
Bake the cookies for 12-14 minutes, or until they just begin to set. Be careful not to overbake; the cracks in the cookies will look slightly undercooked, but they will firm up as they cool on the drying rack, remaining soft and chewy.