Heat oven to 350 degrees. Line 2 large baking sheets with parchment paper.
Melt butter in a skillet or saucepan over medium-low heat until bubbly but not browned, about 3 minutes. Remove saucepan from heat and add chopped chocolate, cocoa powder and espresso over the chocolate mixture. Without stirring, let the mixture sit so the residual heat can melt the chocolate thoroughly while you whip the eggs and sugar. Stir and set aside.
Put the eggs, both sugars and kosher salt in the bowl of a stand mixer fitted with a whisk attachment. (If using a hand mixer, a large bowl will do.) Whisk on medium-high speed until the mixture is pillowy and the sugars have begun to dissolve, 3 to 5 minutes.
Stir the chocolate mixture until glossy and smooth. (If any solid pieces of chocolate remain, you can microwave the mixture in 10-second bursts until everything is melted.)
With the mixer on low speed, add the vanilla extract and then the chocolate mixture. Scrape the sides and bottom of the bowl to make sure the chocolate is evenly distributed, then add the flour and mix on low speed until only a few streaks of flour remain. To avoid overmixing, use a spatula to finish folding in the flour. The dough should be glossy and resemble a very thick brownie batter.
Using a 2-tablespoon/1-ounce scoop, scoop a heaping amount of the dough into mounds directly onto the parchment-lined baking sheets, with each portion at least 2 inches apart, yielding about 18 cookies. Work quickly to ensure the cookies stay shiny once baked.
Bake for 8 minutes until the cookies have started to spread and take on a shiny outer surface, then remove the pans from the oven and whack them on the countertop a couple times to create a cragged top. (This also helps create a fudgier consistency.) Top with flaky sea salt and return to the oven to finish baking, for another 2 minutes until shiny and slightly puffed. Cool for a couple minutes directly on the baking sheets before transferring to a wire rack to cool completely.