In a medium saucepan, melt the butter over medium-high heat. Swirl the pan gently as it cooks until the particles start to turn golden brown and the butter emits a nutty aroma, which should take about 5 minutes.

Remove the pan from heat and continue swirling for another 15 seconds until the butter reaches a rich brown color. Pour the butter into a medium bowl, add the ice cube, and mix well. Place the bowl in the refrigerator to cool completely, which will take about 20 minutes, whisking occasionally. Alternatively, you can whisk the mixture over an ice bath to speed up the cooling process.

While the butter cools, combine the flour, baking soda, and salt in a large bowl and whisk together.

In the bowl of a stand mixer fitted with the whisk attachment, combine the granulated sugar, eggs, and vanilla extract. Whisk on medium-high speed until the mixture becomes a pale brownish-yellow and falls from the whisk in thick ribbons, which should take about 5 minutes.

Switch to the paddle attachment on the mixer. Once the brown butter mixture has cooled and is starting to turn opaque, add the brown sugar and the cooled brown butter to the egg mixture. Mix on medium speed until combined, about 15 seconds.

Add the flour mixture to the bowl and mix on low speed until just combined, leaving some dry flour visible, which should take about 15 seconds. Then, add the chopped chocolate and mix on low speed until the dough comes together, about 15 seconds more.

Transfer the dough to an airtight container and refrigerate for at least overnight, or up to 3 days.

When you're ready to bake, adjust the oven racks to the upper and lower-middle positions and preheat the oven to 325°F (160°C).

Using a 1-ounce portion scoop or a spoon, place scoops of cookie dough onto a nonstick or parchment-lined baking sheet. Each scoop should be about 3 tablespoons in volume, fitting 6 to 8 balls per sheet. Tear each ball in half to expose a rough surface, then reassemble them with the rough sides facing outward.

Bake the cookies in the oven until the edges are golden brown but the centers remain soft, which should take about 13 to 16 minutes. Rotate the pans halfway through baking.

Once baked, remove the sheets from the oven. While the cookies are still warm, lightly sprinkle them with coarse salt and gently press the salt into the cookies. Allow them to cool for 2 minutes before transferring to a wire rack to cool completely.

Repeat the baking process with the remaining cookie dough. Store the cooled cookies in an airtight container, plastic bag, or cookie jar at room temperature for up to 5 days.