Preheat the oven to 375 degrees F. Line baking sheets with parchment paper.
For the rum cookies: Using an electric mixer cream the butter, sugar, vanilla, and rum extract together until light and fluffy—3-5 minutes. Add the egg and beat until combined.
Add the baking powder and salt, then slowly add the flour to the mixture until well combined. Chill the dough for at least 30 minutes to firm up.
Scoop small 1- to 1 1/2-teaspoon portions of dough and roll into balls. Place a little extra sugar in a small bowl and roll each ball in sugar before placing on a baking sheet. Use the bottom of a glass to press the dough balls into flat disks 1/2-inch thick. Bake for 9-10 minutes, until just baked through.
For the Mascarpone Espresso Filling:
Wash the mixing bowl and dry well. Then place the mascarpone and butter in the bowl and beat to combine. Pour the vanilla extract into a small bowl. Add the instant espresso powder and mix to dissolve. Beat into the cheese mixture. Slowly add the 2 cups powdered sugar until light and fluffy. Add extra powdered sugar if needed.
Place the filling in a large zip bag and cut a tiny hole off the corner. Turn half of the cooled cookies over and pipe 1 teaspoon of filling onto the bottom cookies. Top each cookie sandwich and sprinkle with cocoa powder.