Melt butter in a medium saucepan over medium-high heat.
Brown butter by gently swirling pan constantly over medium-high heat, until particles begin to turn golden brown and the butter smells nutty, about 5 minutes.
Remove butter from heat and continue swirling the pan until the butter is a rich brown, about 15 seconds longer.
Transfer to a medium bowl and whisk in ice cube until it melts completely.
Transfer butter to refrigerator, and allow to cool completely, about 20 minutes, whisking occasionally.
Whisk together flour, baking soda, and salt in a large bowl.
Add granulated sugar, eggs, and vanilla extract in the bowl of a stand mixer. Using the whisk attachment, beat on medium high speed until mixture is pale brownish-yellow and falls off the whisk in thick ribbons when lifted. This should take about 5 minutes.
Switch to use the paddle attachment on the mixer. Add brown sugar and cooled brown butter (mixture will have started to firm up at edges and turn opaque) to egg mixture in stand mixer. Mix on medium speed to combine, about 15 seconds.
Add flour mixture and mix on low speed until just barely combined but some dry flour still remains, about 15 seconds.
Add chocolate and mix on low until dough comes together, about 15 seconds longer.
Transfer dough to an airtight container and refrigerate at least overnight and up to three days.
When ready to bake the cookies, adjust oven racks to upper and lower middle positions and preheat oven to 325°F.
Using a 1-ounce ice cream scoop or a spoon, place scoops of cookie dough onto a non-stick baking mat or parchment-lined baking sheet. Each ball should measure approximately 3 tablespoons in volume. This recipe yields approximately 28 cookies. and you should be able to fit 6 to 8 balls on each sheet.
Transfer to oven in batches of 6 to 8 per sheet. Rotate pans back to front and top and bottom halfway through baking.
Bake until golden brown around edges but still soft, 13 to 16 minutes,
Remove baking sheets from oven. While cookies are still hot, sprinkle very lightly with salt, pressing gently into tops.
Let cool for 2 minutes, then transfer cookies to a wire rack to cool completely.