The closest thing to William Shirer is probably David Remnick's Lenin's Tomb, as in both are works by American journalists living in-country as correspondents in the specific years that they cover.

Of course you won't get something exactly like that for the entirety of Soviet history, because that lasted for about 75 years. But if you were going to go for an introductory history for the entire period, you could try Ronald Grigor Suny's The Soviet Experiment, specifically the 2011 edition. This covers Russian tsarist history in the decades just before 1917 and Russian post-Soviet history as well, so it's pretty thorough. Suny is a respected Soviet historian, and each chapter has a helpful "Further Reading" if you want to dive deeper into any particular period or topic. The one big caveat is that he wrote it without any footnotes or endnotes.