Preheat the oven to 350° F. Line 2 baking sheets with parchment paper. In a mixing bowl, beat together the butter, brown sugar, espresso, and vanilla until light and fluffy, about 3-5 minutes. Add the egg, mixing to combine. Then, add the molasses and mix to combine. Add the flour, baking soda, ginger, cinnamon, and salt, beating until combined. Place the sugar in a small bowl. For drop cookies: Roll the dough into tablespoon-size balls (if the dough is too sticky, add 2-4 additional tablespoons flour), then roll through the sugar. Place on the prepared baking sheet, spacing the cookies 2 inches apart. Bake for 8-10 minutes or until the cookies are just starting to set around the edges, the centers should be a little doughy. Let cool on the pan. For cutouts: Divide the dough in half. Roll out the dough on a floured piece of parchment paper to 1/4 inch thickness. Make sure you're using enough flour or your dough will stick. Cut out the cookies into desired shapes. Transfer the cookies to a parchment-lined baking sheet. I recommend using a floured spatula to lift the cookies. Cover the baking sheet and place in the freezer until firm, 15 minutes. Roll out the leftover scraps, and repeat with the remaining dough. Bake for 8-10 minutes, until just set.Meanwhile, make the icing. Add the butter to a pot set over medium heat. Allow the butter to brown lightly until it smells toasted, about 2-3 minutes. Remove from the heat. Whisk in the powdered sugar, vanilla, a pinch of cinnamon, and salt. Immediately spread the icing over the cookies, it will set quickly. Store in an airtight container for up to 5 days.