Making the espresso syrup: Into a bowl, pour the hot espresso, sugar, and rum. Stir until dissolved.
Making the cupcake batter: In the large bowl of a stand mixer with the whisk attachment, beat the eggs with the sugar and rum until thick and pale and tripled in volume, about 8 minutes. In another large bowl, sift together the flour and salt using a wire-mesh sieve. Delicately fold one-third of the flour mixture into the egg mixture. Repeat two more times, until all the flour is incorporated into the egg mixture. Delicately fold in the melted butter.
Preheat the oven to 350 degrees F. Line and butter a 12-cup muffin tin.
Making the cupcakes: Divide the batter equally among the prepared muffin cups. Bake on the center rack until the cupcakes are slightly golden around the edges and a toothpick comes out clean, 15 to 20 minutes. Let cool on a rack for 10 minutes.
Making the icing: In a medium bowl, combine the mascarpone, icing sugar and 3 tablespoons espresso syrup. Beat with a hand mixer until smooth. Set aside.
Brush each cooled cupcake generously with the remaining espresso syrup. Using a pastry bag or butter knife, ice each cupcake with 1/4 cup icing. Garnish with dark chocolate shavings. Refrigerate for 1 hour, or enjoy right away.