Preheat oven to 375 F. Grease a large cast iron skillet (mine is 12”) with cooking spray. In a small bowl, combine flour and salt, then set aside.
In a medium saucepan, add butter and melt over medium heat until butter is brown and nutty, about 4-5 minutes, swirling the pan every minute or so.
Place brown sugar in the bowl of a stand mixer with paddle attachment fitted. Pour browned butter on top, then mix until just combined. Let sit until butter has cooled completely, about 30 minutes.
Once cool, beat the butter and brown sugar on medium high speed until light and fluffy, about 2 minutes, scraping down the sides as needed.
Add eggs, one at a time, beating after each addition, then add vanilla. Mix until just incorporated, then scrape down the sides again.
Add the flour to the stand mixer in 3 batches, mixing in each addition for about 5 seconds, and then scraping down the sides. The flour should just barely be mixed in. Pour cookie dough into prepared cast-iron pan, then smooth the top with a spatula.
Heat cookie butter in a microwave safe bowl for about 20-30 seconds, just to loosen it up a bit. Spoon about ½ cup of the cookie butter over the top of the cookie, then use the back of a knife to swirl it all around. Reserve the remaining ¼ cup of cookie butter.
Bake until cookie is browned around the edges, but still a little soft in the center, about 18 minutes (I used a 12" skillet — might take longer if using a smaller skillet as batter will be thicker!).
Drizzle remaining cookie butter (reheat if necessary)all over the top, then mount the shaved chocolate directly in the center. Cut into wedges or eat directly out of skillet with spoons!