At the moment, the attack is not yet practical because it requires access to millions and possibly billions of copies of the same data encrypted using different keys. A browser would have to make that many connections to a server to give the attacker enough data. A possible exploitation path is to somehow instrument the browser to make a large number of connections, while a man in the middle is observing and recording the traffic.
Does it need to be a single browser? If you have recorded a substantial portion of internet traffic (as the NSA has), and thus you have connections from a bunch of different browsers to a popular SSL/https web page, and the server (during a given time period) returns the exact same response (it's static web page) to all of them (or a portion of them and you can figure out which portion)....
?
I can think of a bunch of different reasons this would NOT work, but (or 'and') I'm also not neccesarily an expert and don't entirely understand this stuff, so I'm not sure either way, just wondering.