Preheat oven to 350°F Line 2 cookie sheets with parchment or silpats.
In a small skillet, melt 10 tbsp unsalted butter. Continue to cook until the milk solids begin to brown, 2-4 minutes. Transfer to a bowl, add the remaining 4 tbsp of cold butter, and set aside to cool, 15 minutes.
In a pie dish, combine 1/4 c packed brown sugar with white sugar, rubbing with fingers to break up the brown sugar. Set aside.
In a large bowl, whisk together flour, baking soda and baking powder (the soda counteracts the acid in the brown sugar, while the powder provides lift; you really need both!).
In a large bowl, combine remaining 1 3/4 c brown sugar with cooled melted butter, along with vanilla and salt. Stir to combine until smooth.
In a small bowl, briefly beat egg and egg yolk to combine. Add to butter and sugar and stir well. Add flour mixture and stir until well-mixed and no pockets of dry flour remain.
Turn dough onto a cutting board and pat into a rough circle. Cut circle in four equal pieces, then divide each piece into six equal pieces, for a total of 24 cookies. Quickly roll each lump of dough into a ball, then roll in the mixed sugars to coat, then place onto lined cookie sheets, about 2" apart.
Bake ONE TRAY AT A TIME for 12-14 minutes, spinning the tray 180° halfway done. (You need the top and bottom to cook evenly, or the texture will be wrong; if you have both pans in at the same time, they will shield each other.) They will appear underdone, but the edge should be set enough that you can lift it, and when pressed gently halfway between the center and the edge, the cookie should spring back. Let sit on the hot tray 5 minutes to finish cooking, then transfer to a rack to cool completely.