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To brown the butter: In a small, deep pot over medium-high heat, cook the butter, whisking frequently to scrape the bottom of the pot, until it’s a light honey color and little brown bits form on the bottom of the pot, 4 to 6 minutes. Remove the pot from the heat and allow the butter to cool slightly for 5 minutes. You should have a scant 1/2 cup (85g) of brown butter.
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Transfer the brown butter to a small bowl or container, cover, and refrigerate until it’s opaque and creamy in consistency, about 1 hour.
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Once the butter has chilled, remove it from the fridge.
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To make the dough: In a large mixing bowl or bowl of a stand mixer, combine the sugars and brown butter. Mix until it is evenly combined and the mixture has a wet, sandy appearance, about 3 minutes (or a bit longer if mixing by hand).
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Add the starter, egg, and vanilla, mixing to fully incorporate.
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Add the flours, salt, baking powder, and baking soda, mixing just until the ingredients are incorporated.
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Switch to a spatula and add 1 cup (170g) of chopped dark and 1 cup (170g) of chopped milk chocolate, mixing to fully combine. The remaining 1/4 cup (43g) of each chocolate will be used for topping the cookies.
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Scoop the dough onto a parchment-lined baking sheet into heaping 2 tablespoon portions (57g, 2 ounces); a heaping jumbo cookie scoop is about right. If you’re scooping by hand, they should be about the size of golf balls. The dough balls can be placed close together so that they fit on a single baking sheet. Cover and refrigerate overnight, or at least 8 hours.
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Preheat the oven to 350°F with two racks in the middle third of the oven.
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Remove the baking sheet with the scooped dough from the fridge. Combine the reserved chopped chocolate in a small bowl.
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One at a time, roll each scoop of cookie dough into a neater ball in the palms of your hands, then dip it into the chocolate so that several pieces adhere to the top.
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Arrange the cookies on parchment-lined baking sheets, spacing them 3" apart, with no more than 6 cookies per tray.
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Bake the cookies for 18 to 22 minutes, until set and slightly golden around the edges but soft in the middle.
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Remove from the oven and, while still hot, sprinkle each cookie with a little flaky sea salt, if desired. Repeat with the remaining dough until all the cookies are baked. Allow them to cool for at least 10 minutes on the sheet pans before enjoying or transferring to a wire rack to cool completely.
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Storage instructions: Store baked sourdough chocolate chip cookies at room temperature in an airtight container for up to 5 days. After rolling the portioned dough in the chopped chocolate, wrap well and freeze for up to 2 weeks. The cookies can be baked directly from frozen the same way, for the full 22 minutes, although they may not spread out as much as fresh, unfrozen dough.
