In a saucepan, melt the butter over medium heat. Continue to cook the butter, swirling occasionally, until the water evaporates, and the butter forms fragrant, nutty, golden brown solids at the bottom of the pan. Remove from the heat and immediately pour the hot butter into a bowl, scraping all the browned bits off the bottom of the saucepan with a spatula.
Let the butter cool at room temperature for several hours, until it becomes somewhat solid again. As the butter cools, it may not become firm enough to hold its shape (depending on how warm your kitchen is), and may remain a little soft and spreadable, but allow it to cool until it is no longer a liquid.
In the bowl of your stand mixer fitted with the paddle attachment, beat the cooled browned butter, granulated sugar and brown sugar for 5 minutes on medium speed, scraping the bowl occasionally, until very light and fluffy. Gently mix in the egg and the vanilla.
In a separate bowl, combine the flour, baking soda, espresso powder, 3/4 teaspoon salt and the chopped chocolate. With the mixer on low, add the dry ingredients and mix just until combined, briefly increasing the speed to medium to incorporate all of the flour.
Use a cupcake/ice cream scoop with a release lever to divide the dough into 17 portions, about 2 ounces each. Shape the dough into balls, place in an airtight container, and refrigerate for at least 2 hours, or overnight.
Preheat the oven to 350 F, and position a rack in the center of the oven. Line a large baking sheet with parchment paper. While the oven is preheating, place 8 balls of dough on a plate in the freezer for 10 minutes.
Place the frozen balls of dough on the baking sheet, spaced three inches apart. Bake for about 13 minutes, until the edges are crisp and light golden brown, and the centers are gently puffed and look slightly underdone. If your cookies didn't spread quite enough, you can bang the pan on the counter several times to help them flatten and settle.Sprinkle the cookies with the remaining 1/4 teaspoon salt. Let cool on the pan for 1 minute, then transfer to a wire rack to cool completely. As the cookies cool, the soft centers will settle down a bit, leaving a lightly crisp edge around the cookies with chewy centers.TIP: For perfectly round cookies, immediately after taking the pan out of the oven, use a round cookie cutter that's larger than the circumference of the baked cookies, and "scoot" it around the hot cookies. This pulls in the uneven edges and makes beautifully round cookies. You have to do this quickly, before the cookie sets. Repeat the above steps to freeze and bake the other 9 balls of dough. Store cooled cookies in an airtight container at room temperature for several days, or in the freezer for up to 3-6 months.