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To brown the butter: Place half the butter (113g) in a medium skillet and melt it over medium heat, swirling the pan occasionally. The butter will foam and froth as it cooks and start to crackle and pop. Once the crackling stops, keep a close eye on the melted butter, continuing to swirl the pan at intervals. The butter will become fragrant, and brown bits will form on the bottom. Once the bits are amber brown (about 2 1/2 to 3 minutes or so after the sizzling stops), remove the butter from the burner and pour it into a small bowl, brown bits and all. Allow it to cool for 20 minutes.
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To make the dough: In a large bowl or the bowl of a stand mixer, beat the remaining 114g of butter with the brown sugar for 3 to 5 minutes, until the mixture is very smooth. Beat in the vanilla.
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Pour the cooled brown butter into the bowl along with the granulated sugar. Beat for 2 minutes, until smooth; the mixture will lighten in color and become fluffy.
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Add the eggs and beat for another minute.
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Add the flour, salt, baking soda, and cardamom, beating on low speed just until everything is thoroughly combined.
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Use a spatula to fold in the chopped chocolate.
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Scoop the dough into 12 equal pieces (about 90g each); a muffin scoop works well here. Place onto a piece of parchment paper, waxed paper, or plastic wrap, cover, and refrigerate for at least one hour (until firm), or up to two days.
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To bake the cookies: About 15 minutes before you're ready to begin baking, preheat your oven to 375°F with racks in the center and upper third. Lightly grease (or line with parchment) three baking sheets. If you don’t have three pans, plan to bake in batches.
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Arrange four cookies on each pan, leaving as much room as possible between them. Sprinkle the cookies with coarse salt
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Bake the cookies for 10 to 12 minutes, until the edges are just starting to turn golden brown. Remove them from the oven and allow them to rest on the pans for at least 5 minutes before moving them. They can cool completely right on the pans, or transfer them to a rack to cool more quickly.
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Storage information: Store cookies, well wrapped, at room temperature for several days; freeze for longer storage. (See "tips," below, for details on freezing.)