Maybe if you use a third party DNS service, but then you need to trust them.
edit: You're still broadcasting the IP you're connecting to, but it's still nice to close up this DNS lookup leak.
This is all the more reason we need to start encrypting all communication. All my hand built services (home bots, etc) need to start using tls for everything.
I know very little about the nitty gritty of HTTPS, so forgive my ignorance, but i thought the most i could do was try to pass off a custom key (ie, spoof the key authority), but then the signing done from the https site (say, https://google.com) wouldn't be valid based on my bad key.
How common is it to read full https data if you're a MITM?
STARTTLS on mail is a slightly different story, though I'm going to assume that most of the established compaines are smart enough fo avoid email snooping.
You might, however, be surprised at how much you give away via the metadata associated with your web browsing.