Line 2 baking sheets with parchment paper, set aside.
In a medium bowl whisk together the flour, baking soda, baking powder, salt, and espresso powder. Set aside.
In a large bowl use a stand mixer with a paddle attachment or a hand mixer to beat the brown sugar, granulated sugar, and butter together until smooth.
Add in the eggs, vanilla extract, and rum extract until combined, scraping down the sides of the mixing bowl as needed.
Add in the dry ingredients until incorporated.
Use a 2-tablespoon cookie scoop to scoop cookie dough onto the prepared baking sheets, fitting about 6-8 cookies per sheet. Press the cookie dough down slightly to flatten the tops.
Bake in the preheated oven for 11-13 minutes until the cookies are golden on the edges. Cool completely.
Meanwhile, beat the mascarpone, heavy cream, powdered sugar, and vanilla extract with a stand mixer or hand mixer until stiff peaks form.
Spread or pipe the frosting on top of the cooled cookies.