How can an EB-5 conditional green card holder re-enter the U.S. with an expired extension?

You have to go to the consulate and explain why you have been out and try and get a boarding foil. Or you can present yourself at a land border post and explain to the officer they have discretion to admit you since you are a lawful permanent resident.

You can either try booking your flight and boarding your plane as long as you have the original I-797 notice, or depending on your airline, you can obtain an I-131A carrier document to allow you to reenter the U.S.

Not a great situation. Have you been out of the country for a long time? CBP will probably let you in but the airlines are not likely to let you board. If you can get to a land border crossing that might be your best bet. But, as usual, talk to your attorney. An SB1 is an option (albeit not a great one).

I believe you will need to file an I-131A with the consulate.

You will probably need a boarding foil. You may also want to consider a mandamus, a case pending for two years has been pending for an illegally long amount of time.