One helpful commenter – Phil from Oak Park – provided me with a link to a HuffPo article and a video about how to eat this hamburger, which I found most helpful.
With this approach, you're basically keeping three middle fingers from both hands on the crown of the bun; thumb and pinky go on the bottom, providing uniform pressure on top and support on the bottom.
Last week at Compass Bar, a gastro-pub on Chicago's northside, I ordered the messiest burger I could find, plump with blue cheese, red onion, lettuce and, of course, tomato (keeping this moist disk in place would prove the real acid test).
It seems awkward at first, and honestly after you're half-way through, as the hamburger gets smaller, it becomes difficult to deploy this handling strategy, but damn, it works. By slightly moving the fingers, you can readjust the sandwich so nothing falls out while you're eating.
A side-benefit: it strengthens your pinkies.
Watch the video and try it yourself!