+ 1 to Athens or something and then buy a short flight if there is no availability. But its definitely worth checking. Either way I would use United.com and try to route out a great trip in first class. Use the oneway option to try to piece it together. If there is only partial availability in First (mixed cabin (part of the flight is in economy)) check for later flights from that destination.
It can't hurt to call as well... At first they will say I dont see anything, but if you give them the first leg, "i see flights from IAD-FRA on Lufthansa First", they might get helpful and search the rest of the flight for you.
No pressure as any ticket can be held for 3 days. Call, if you book a route put it on hold and transfer miles in. They transfer instantly from partners like Chase.
If you transfer 109k from Chase to United and another 21k (+5k bonus), you'll have enough for a round trip ticket in first will united miles.

btw you get a stopover and an open jaw... So if you find availability to and not from, you could get that cheap ticket to Athens or wherever and continue your flight.

Also, if thats too much of a headache there are a few people out there that charge a little fee for using your miles, who are very good at what they do.