Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Melt the butter in a medium skillet over medium heat; cook, swirling the pan occasionally, until the butter is light brown, 8 to 10 minutes. Pour into a large bowl and let cool to room temperature, 20 minutes.
Whisk the flour, baking soda and salt in a medium bowl. Whisk the superfine sugar and vanilla into the browned butter. Whisk in the eggs. Stir in the flour mixture, then stir in all the chocolate chips.
Drop tablespoonfuls of dough about 3 inches apart onto the prepared pans. Sprinkle generously with coarse sugar.
Bake, switching the pans halfway through, until the cookies are a deep golden brown, 18 to 20 minutes. Let cool 5 minutes on the pans, then transfer to racks to cool completely.