This is something I ate quite frequently as a kid, and continue to enjoy immensely. I think it is good for a snack, an appetizer, quick breakfast, or served along side rice (make it saltier in that case). At the core, chawanmushi is incredibly simple, as the three-ingredient list shows, but it can be made more complex also.
(1) Beat eggs in a bowl thoroughly but gently. Try to minimize the bubbles.
(2) Add water and soy sauce. In terms of volume, I guess a 6:1:1 ratio of egg:water:soy sauce sounds reasonable to me, but adjust for your preferred taste and texture. Mix.
(3) Cover the bowl with a plate and place it in a microwave for 3~4 minutes.
(4) Serve while hot.
Instead of water, you can use stock or mushroom-rehydration liquid. You can also add other things to the egg mixture, such as mushroom, fishcake, seafood, bamboo, cooked chicken cubes, udon... I also like a bit of pepper and adding a drop of cooking oil to the mixture to give a surface sheen.
To achieve a better texture, you should really use a rice cooker / steamer hybrid. [An Example]
You can also put it in Asian-ish tea cups instead for individual servings, or a bigger ceramic container for bigger portions. Google-image 'chawanmushi' for pictures.
The will be pieces of egginess stuck to the bowl. Rather than trying to scrub them out by brute force, it is easier to get rid of them by leaving the bowl in hot water so that they soften and come off easily.