Cream butter, granulated sugar, light brown sugar, and vanilla extract over medium-high speed until light (pale yellow) and fluffy, about 3 minutes. Scrape down the sides of the bowl as needed.
On low speed, slowly add flour and chocolate. Mix until just combined.
Place the dough on a clean work surface and divide it in half.
Wrap each half in plastic and roll it on the counter to form a log. It should be about 6 inches long and 2 inches in diameter.
Refrigerate the dough for at least 2 hours or until firm. You can also freeze it for about 30 minutes.
Brush the egg on the outside of the log and roll it in turbinado sugar.
Using a sharp knife, carefully slice the log into ½-inch thick rounds. Press them back together if they break apart.
Arrange them about an inch apart on a baking sheet lined with parchment paper. You will need a baking sheet for each log. Sprinkle them with flaky salt.
Bake in a preheated oven of 350° F (177° C) for 12 to 14 minutes until the edges start to brown. Let them cool on the baking sheet.