Prepare the baking sheets. Preheat the oven to 350°F. Line 2 cookie sheets with parchment paper and set them aside.
Whisk together the eggs and sugars. In a stand mixer with a whisk attachment, combine the eggs and both sugars. Beat on high speed for 7 minutes. The mixture will be thick, doubled in volume, and have a pale light brown color.
Melt the chocolate and butter. In a microwave-safe bowl, combine the butter and finely chopped chocolate. Microwave in 15-second increments, stirring well between each until the chocolate is completely smooth and melted. Do not overheat the chocolate as this can burn it. Set aside to slightly cool.
Add the melted chocolate to the whipped eggs. With the mixer on low, pour in the melted chocolate/butter mixture along with the vanilla extract. Mix until combined for 30-45 seconds, making sure to scrape down the sides.
Add the dry ingredients. Sift in the gluten-free flour, cocoa powder, salt, espresso powder and baking powder. Mix until just combined, then turn off the mixture. Using a spatula, scrape the bottom of the sides of the bowl and fold in any remaining flour bits. The dough will be glossy and soft, similar to brownie batter. You'll want to work quickly to get the distinct glossy crackly tops, do not let the dough sit!
Scoop and bake. Use a large #20 cookie scoop (around 3 tablespoons) to scoop the cookie dough onto the prepared baking sheets, about two inches apart. Bake one tray at a time for 9-10 minutes, or until the edges are set, but the center is still gooey. Remove the cookies from the oven and sprinkle the tops with flaky sea salt. Round out the cookies. To achieve perfectly round cookies, place a large cookie cutter or glass cup slightly bigger than the cookies over the top of the cookies, and carefully swirl in a circular motion rounding out the edges. Let the cookies sit on the baking tray for 5 minutes before transferring them to a wire rack to finish cooling.