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To make the cookie dough: Coarsely chop the semisweet chocolate; a serrated knife works well here. Set aside 1/4 cup (43g) of the chocolate.
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In a large microwave-safe bowl, melt the remaining 1/4 cup (43g) chocolate in 20-second increments, stirring between each burst, until smooth. Immediately add the butter and stir to coat the pieces in the warm chocolate (this will help the butter begin to soften). Add the brown sugar and smash the mixture together with a flexible spatula until the butter begins to break down. Switch to a whisk and beat vigorously to break up all of the butter lumps and dissolve the sugar, about 2 minutes; the mixture will become noticeably airy and pale. (See “tips,” below, for instructions on using a stand mixer.)
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Whisk in the whole egg, followed by the yolk, stirring until the mixture is creamy and mousse-like. Whisk in the vanilla, baking powder, salt, baking soda, and espresso powder.
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Return to the flexible spatula and stir in the flour and cocoa powder until almost no dry streaks remain. Add the reserved chopped chocolate and stir until evenly incorporated.
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Scrape the sides of the bowl, cover tightly, and chill in the refrigerator for at least 2 hours, or up to overnight, until the dough is firm but scoopable.
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When ready to bake, preheat the oven to 350°F with a rack in the center. Line a baking sheet with parchment.
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To prepare the sugar coating: In a shallow bowl, combine the sugar and salt.
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With a spoon or a leveled Tablespoon Cookie Scoop, scoop a few balls (about 27g each or a generous 1 tablespoon) of the soft dough directly into the bowl with the sugar coating. Roll each in the sugar to make a rounder ball (no need to make perfect spheres) and place on the prepared baking sheets, eight cookies per sheet, spacing them evenly apart.
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Bake the cookies, one sheet at a time, for 8 to 11 minutes, just until the edges are set; the centers should still be soft but not look wet. Let the cookies cool on the baking sheet for at least 5 minutes before carefully transferring them to a wire rack to cool completely. Repeat with the rest of the cookie dough.
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To make the cookie frosting: Using a hand or stand mixer, mix the confectioners’ sugar and butter until just combined. With the mixer on medium speed, gradually add the vanilla and the desired amount of water or milk. (Less liquid will yield a stiffer frosting that will dry slightly and make cookies that are easier to stack; more will result in a softer, buttercream-like consistency that stays soft for several days.) Beat on high speed until fluffy, 1 to 2 minutes.
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Spread about 2 teaspoons (10g) frosting onto each cooled cookie.
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Storage information: Store the chocolate sugar cookies airtight at room temperature for up to 5 days. Freeze for longer-term storage.
