In a glass measuring cup, melt the butter and place the espresso grounds in the cup and stir to dissolve the coffee.
Place the cup in the refrigerator to chill into a solid again.
Once the butter has turned back to solid, preheat the oven to 375 degrees.
In a small bowl, whisk the flour, salt, cornstarch, baking powder, and baking soda.
In a large bowl, beat the cooled, espresso butter and sugar until light and fluffy.
Add in the egg, vanilla, rum extract, and egg and beat until creamy.
Add in the flour mixture until the flour is completely combined.
Use a cookie scoop to make round cookies about 1 ½” tbsp in size.
Place on a baking tray 3 inches apart and bake for 10-12 minutes, until the edges start to turn golden.
Remove and allow to cool completely before frosting.
To make the frosting, beat the mascarpone cheese, espresso, vanilla and rum extract until creamy.
Start by adding 2 cups of powdered sugar and beat until smooth.
Add in the heavy cream and continue to beat until smooth and creamy.
**You may add more powdered sugar to make a thicker frosting if desired. Adjust as wanted.
Place the frosting in a piping bag with a round tip and pipe circles of frosting on top of each cooled cookie.
Sprinkle the tops with cocoa powder.