I finally found a solution (thanks to Google, I just had to keep trying different combinations of search terms). Thanks to the comment from arr1 on this page (it's about two thirds of the way down the page).
I have yet to actually test this but, in short, you send two headers: one that tells the browser exactly how much data to expect then one to tell the browser to close the connection (which it will only do after receiving the expected amount of content). I haven't tested this yet.
