Add 1 stick butter to a small saucepan set over medium heat, cooking until the butter begins to brown, about 3-4 minutes. Remove from the heat and transfer to a heatproof bowl. Stick in the freezer to chill, 10-15 minutes, but no longer.In a large mixing bowl, beat together the remaining stick of butter, the cooled browned butter, brown sugar, and vanilla until creamy. Add the flour and mix until fully combined. If the dough is still too sticky to form a ball, add 1/4 cup flour. Stir in the pecans.Place the dough on a large piece of wax paper or plastic wrap. Using your hands, shape it into a log about 12 inches long and about 2 inches in diameter. Wrap the dough up in the wax paper and place in the fridge for 2 hours or up to 5 days.Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper.Unwrap the dough and place it on a cutting board. Brush the dough with the beaten egg and roll in coarse sugar to coat. Using a sharp chef's knife, cut the log into 1/2 inch slices. It is important to cut in a quick forward motion. Do not move the knife back and forth. If the dough cracks, just push the dough rounds back together. Promise it will be ok. Arrange the slices on the prepared baking sheet. Bake the cookies for 12-14 minutes, until the edges are golden brown. Let the cookies cool on the baking sheet. Drizzle the chocolate over each cookie and sprinkle with flaky sea salt. To harden the chocolate, place the tray of cookies in the fridge for 10 minutes. Store cookies in an airtight container for up 4 days at room temperature.