Line two baking sheets with parchment paper. Set aside and position an oven rack in the center, but wait to preheat.
Brown the butter: In a light-colored saucepan over medium heat, cook the butter, stirring and scraping often, until the milk solids turn deep amber and smell nutty, 5–7 minutes. Immediately scrape all the butter and browned bits into a large mixing bowl. Cool until just warm to the touch, about 10 minutes.
Whisk dry ingredients: In a separate bowl, combine the flour, milk powder, baking soda, fine sea salt, and espresso powder.
Build the base: Add both sugars to the warm brown butter. Whisk vigorously for 30 seconds, rest 2 minutes, and repeat once more until the mixture looks thick and glossy.
Whisk in the egg and vanilla until smooth and slightly lightened, 30–45 seconds.
Fold in the dry ingredients with a spatula just until a few streaks of flour remain. Add most of the chopped chocolate (reserve a small handful) and fold until evenly distributed and no dry flour remains.
Rest the dough at room temperature for 20 minutes to hydrate. Meanwhile, preheat the oven to 360°F (182°C).
Scoop 2-tablespoon portions (about 40g) onto the prepared sheets, spacing 2 inches apart. Press a few shards of the reserved chocolate on top of each mound for puddly tops.
Chill the trays: Refrigerate 15 minutes (or freeze 8 minutes) to help control spread.
Bake one sheet at a time for 10–12 minutes, rotating halfway, until the edges are set and lightly browned and the centers look slightly underdone with a soft gloss.
Cool on the sheet for 5 minutes, then move cookies to a wire rack to finish cooling.