Whisk together the eggs and sugar in a large heatproof bowl or a double boiler if you have one. I don't, so I just use a metal bowl.
Put the bowl over a pot of simmering water.
Whisk constantly until the eggs triple in volume and turn pale yellow. It takes about 5 minutes.
Pour the eggs into the mixer and beat for about 5 minutes. The bowl will feel cool when it is ready.
Add the mascarpone, one tub at a time until it is combined.
Fold in the chocolate chips. Chill for at least one hour.
In a large bowl, stir together the boiling water, brown sugar, espresso powder, cocoa powder, and Kahlua. Stir until everything has dissolved. Set aside and let cool.
Pour the coffee mixture into a shallow dish. Dip the ladyfingers into the mixture, and line the bottom of a 9x13 pan with the cookies. You want the ladyfingers pretty wet, but not so wet that they fall apart. When I was done with my layer, I would brush more of the coffee mixture over each layer just to be sure that they were really soaked.
Layer 1/3 of the mascarpone mixture over the ladyfingers. Repeat the layers until you have 3 layers of each. Alternate the direction of the cookies, it helps the tiramisu hold together when you cut it. Cover and chill for 2 hours. Overnight is even better. You can make this a few days ahead of time.
Cover the top with shaved chocolate before serving.