Our Sunni brothers claim that the Prophet said that the best of his Ummah was the first generation of the Sahabah, followed by the generation after that. Do the Shia also have such a narration?
No. In this day and age, this is basically a Salafi idea which is used to justify the belief that reconstructing society as it was during the first four caliphs will fix all the problems in the world.
Since Shi'is do not share the idea that society was ideal or perfect during the first four caliphates, and in fact point to a number of problems during that time, or things which were done which may have been against the Prophet's intent (whether relating to internal matters, such as the caliphate or financial management, or external matters, such as wars), this idea isn't found in Shi'ism.
In fact, we look forward to better times during the advent of the Mahdi (A)!