In the bowl of a stand mixer fitted with the paddle attachment, beat the cold butter and sugars together until fluffy, about 4 minutes or so. Add the vanilla and beat for 30 seconds.
Add the flour, kosher salt, pecans, and chocolate. Beat until the dough clumps together, about 2 minutes or so. Divide the dough into two equal portions, roll into logs about 2 1/2 inches thick and 6-7 inches long, and wrap tightly with plastic wrap. Chill logs in the fridge for 2 hours.
Preheat an oven to 350 degrees F and line two baking sheets with parchment paper. Brush the outside of the logs with the beaten egg and roll in the coarse sugar until well coated.
Slice the shortbread into rounds just under 1/2" thick with a serrated knife. If the dough breaks apart, just press it back together.
Place the cookies on the prepared baking sheets and bake for 15 minutes, until just barely golden brown. Cool to room temperature and then dip the cooled cookies in the melted chocolate. Sprinkle the top with flaky sea salt and more chopped pecans.