Prepare the filling first. In a medium bowl, beat the softened mascarpone, 1/4 cup powdered sugar, and vanilla extract until perfectly smooth. Use a hand mixer or a whisk. The mixture should look like thick frosting.
Scoop the filling into 12 small portions. Place them on a parchment-lined plate or small baking sheet. The portions should be about the size of a hazelnut. Freeze for at least 30 minutes, or until completely solid. If you freeze them longer, they will be even easier to handle.
Create the espresso mixture. In a small bowl, dissolve the instant espresso powder into the coffee liqueur or warm water. Stir until the granules are completely gone and the liquid looks like a thick syrup. Set this aside to cool slightly.
Cream the butter and sugar. In a large mixing bowl, beat the softened butter and granulated sugar for 2 to 3 minutes. You want the mixture to be light and fluffy. This incorporates air, which helps the cookies bake with a nice texture.
Add the wet ingredients. Beat in the eggs one at a time, mixing well after each addition. Scrape down the sides of the bowl. Then, blend in the dissolved espresso mixture until it is fully integrated. The batter will look a bit silky at this stage.
Mix the dry ingredients. In a separate bowl, whisk together the all-purpose flour, sifted cocoa powder, baking powder, and fine sea salt. Whisking ensures the leavening agent is evenly distributed.
Combine wet and dry. Gradually mix the dry ingredients into the wet ingredients on low speed. Mix just until the dough comes together and no dry streaks remain. Overmixing can make the cookies tough, so stop as soon as it looks uniform.
Chill the dough. Cover the bowl with plastic wrap and refrigerate for at least 1 hour. The dough will be very soft right now. Chilling it solidifies the butter, which prevents the cookies from spreading too much in the oven. It also makes the assembly process much cleaner.
Preheat the oven. When you are ready to bake, preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper. Pour the 1/2 cup of sifted powdered sugar into a shallow bowl. This is the coating station.
Assemble the cookies. Scoop 2 tablespoons of the chilled dough. Flatten it gently in the palm of your hand to create a disc. Place one frozen mascarpone ball in the center. Wrap the dough completely around the ball, pinching the edges tightly to seal. Roll it between your palms until it is a smooth sphere.
Coat in sugar. Roll the stuffed dough ball generously in the powdered sugar. You want a thick, opaque layer of sugar to cover every inch. This coating is what creates the iconic crackle effect. Place the ball on the prepared baking sheet.
Bake. Repeat for the remaining cookies, spacing them about 2 inches apart. Bake for 11 to 13 minutes. You will see the dramatic crinkles form as the dough expands. The edges should look set, but the centers will still look soft and slightly underdone. That is exactly what you want for a fudgy texture.
Cool and finish. Let the cookies cool on the baking sheet for 5 minutes. This allows them to set up before you move them. Transfer to a wire rack to cool completely. Just before serving, finish with a light dusting of cocoa powder through a fine mesh sieve.