In a metal stand mixer bowl, combine the eggs and granulated sugar. Beat well to combine. Place the bowl on top of a small pot with 1-inch of boiling water; be sure that the bowl does not touch the water. Beat the eggs until they have tripled in size and the whisk leaves a trail, 5 to 7 minutes.
Once the eggs have tripled in size, remove the bowl from the double boiler and put on the stand mixer equipped with the whisk and beat on a medium speed until the bottom of the bowl feels cool, about 5 minutes. Add the mascarpone and chocolate chips and mix until just combined. Refrigerate 1 hour.
Combine the boiling water, brown sugar, espresso powder, cocoa powder, and coffee liqueur. Stir until everything has dissolved. Let cool.
When the espresso mixture has cooled, dip the cookies in the liquid one at a time and line the bottom of a 9 by 13-inch baking dish. Spread 1/3 of the chilled mascarpone mixture over the cookies. Repeat this process 2 more times finishing with a layer of mascarpone. For more stability, lay the second layer of cookies in the opposite direction of the first layer.
Cover and refrigerate for at least a couple of hours and up to overnight; the longer it chills the easier it will be to slice.
Top the tiramisu with finely grated chocolate. Cut into squares and serve.