Twitter – My Twitter timeline has a huge 11 hour period of silence. Can I retrieve those tweets anyhow

The answers so far will only give you at most 100 retweets. There is an API method that returns up to 100 entries of a cursored collection (so you can page through it to get more), though.

There's more information on the Twitter Developers site at GET statuses/retweeters/ids, though you'll also want to read their page about Using cursors to navigate collections.

Edited to add: That said, the first thread on the Developers site that surfaced in a quick google has @episod, a Twitter employee saying:

You can't likely get to all of them. The data is spread out. Your best bet is to use the REST API to determine many of the users who performed the retweet, but the data will still be constrained.

It's easiest to track retweets as they happen rather than try to find them from the past. Use the Streaming API for that.

^{Source: how to get all retweet of a given tweet}

It's not going through your firewall when it talks to the Twitter servers. Brizzly retrieves content from the Twitter API through its own connection and then stores it on their server and presents it to you while you're logged in. (It only does this when you're using it, though, otherwise you'd end up using all of your Twitter API limit.)

If your company doesn't allow usage of Twitter during the work day, I wouldn't recommend using Brizzly to get around it. Also, they may eventually figure it out and block it and similar sites, such as Hootsuite.