Ok I am a big fan myself of Chawanmushi,
the best I have found in Tokyo is at Toriyoshi in Nakameguro (very near Nakameguro Station),
the place is also one of the best places for Yakitori (the chef supposedly learned cooking in France and came back to open this restaurant to get back to his roots), here are the kanji just in case you look for it on the internet: 鳥よし 中目黒店
Now for the down side, the place is only opened at night and briefly at that (it closes at around 10.30PM), it is always packed and does not take reservations, good luck ;)
Which neighborhood in the city? On the Japanese food sites you can throw in specific dishes and it will spit out reviews where those dishes are noted. You have to read the review though to understand what it specifically says. As Robb alludes, it's a ubiquitous dish, so it comes up for a lot of places. It's often served as filler dish at the end of a sushi meal.
Sure, for chawanmushi, there are some Izakaya and sushi set in the air (tune?).
So nothing to add, except a very pleasant lunch set at 1200.-yens with chawanmushi (diner more expensive). Their chawanmushi is a good try at the Kaiseki Akasaka Jotaro(sometimes with Chinese main like schrimp harumaki more Japanese), elegant modern room, ceramics dressing plates. The lunch is limited to 15 lunch seats, so hurry and book it even 10 mn before 11:30!
it worth a try for lunch even it is difficult to find the place.