A bit random, but the Cafe International in Woodley Park makes a mean breakfast sandwich. Real eggs folded over themselves and any other ingredients (meat, cheese) on a bagel, toast, english muffin, croissant. The bagels themselves aren't spectacular, but passable. One of the owners is a bit surly, but most of the staff is very nice.
Bagels Etc. on P Street in Dupont is another solid option. They also have tons of other schmear and sandwich options. Very lunch-counter style, efficient and cheap. And the bagels themselves are denser and chewier, not the puffy "bakery style" generic types.