edit: Also, there is the original Carm's in Hillside on Wolf about 4 blocks north of there. Haven't been in a while, so not 100% certain they have pepper & eggs, but it seems like a safe bet.
Id say a quick call would be worth it, just to check.. but Id be surprised if they didnt have a Pepper-n-egg sandwich at Carm's on a Friday (especially during Lent!)
And, if they do, it'll probably be very good. Ive been to Carm's only a couple of times, and never on a Friday (so didnt get to try their Pepper-n-egg).. but its a fine beefstand, one of the better ones. Their beef and their sausage were both top-tier IMHO - second only to Johnnie's, probably. I would guess their pepper-n-egg sandwich (the Third Wing of the Chicago Beef-stand Holy Trinity) is stellar as well.