"Mekhleme" ($11), the menu tells us, is an "Iraqi 'corned beef' hash" that also includes tomato, onions, and potatoes, all topped by two poached eggs. The Middle East is no stranger to hash, but even when it's the meaty sort, generally hash has been the butt of a joke. (An older generation has similar feelings about creamed chipped beef on toast.)
All online citations that I've found, however, connect "mekhleme" only with Busboys and Poets, generally with favorable notices. (Add mine.) But as to the origin and significance of the name, and as to whether mekhleme is a traditional Iraqi dish or a more recent Iraqi-American invention, I still can't say. (Update: See the comments below.)
Busboys and Poets
1025 5th St. N.W. (K-L Sts.), Washington, D.C.
(one of several locations)
(From an Easter 2013 visit)