Greetings from Beijing!
It depends what your after but heres a few suggestions for beijing in no particuliar order:
Peking Duck - Quan ju De, a little expensive but the duck is good, 14 outlets in beijing.
Churrascos - Crown Plaza Parkveiw Wuzhou beijing, excellant brazillian chrrascos!
Morels - Just get the fresh mussels - good belgian beer.
Vegetarian - Vanilla Garden - great chinese vegetarian in a little hutong house
Green Tea House - Contempory and funky tea infused cusine
Hot Pot - Ding Ding Xiang
if you have anything specific you want to try let me know, and have fun!
For Beijing, I add another 3 recommendation:
-South Silk Road at Shi Sha Hai, Yunnan cuisine,
-Made in China at Grand Hyatt, northern Chinese (Beijing) food,
-CourtYard at 95 Donghuamen Ave, fine dining overlooking Forbidden City.
For Shanghai, please read my recommendation on earlier thread "Shanghai quick eats?":
For Beijing, the Quanjude Duck restaurant is a must, not only for the food but the presentation and the enormity of the place. It can seat 2,500. Out near the Ming tombs, the Friendship Restaurant was pretty good.
In Shanghai, Mei Long Zhen on the Nanjing Road is incredible. When the taxi drops you off, you have to look back into an alcove to find it as it is not streetfront. Amazing menu. For a vegetarian experience, I was impressed by the Gong De Lin Buddhist Vegetarian Restaurant and their very creative imitations of meat and seafood dishes. Lunch at Shanghai mansions was also good.
Bon appetit and voyage!