The thinner hose is the one the cracked and broke off. i was wondering if anyone knows what part/ part number that is in the picture?

 

EGR solenoid, oem replacements are no longer made IIRC but check maybe Classic Industries. You can use C4 Corvette ones which are not cheap or I believe mid 90's Cadillac ones if you want to replace the pig tail as well. If you do a search for EGR solenoids there are a few good write ups on it. If just the hose cracked/broke you can join the 2 broken pieces together with the correct size vacuum hose, will be a whole lot less money.

 

DORMAN 47418 Tubing Dia.: 5/32"