if you have a drill and some bits or and/or a welder this is an easy repair.
the easy way:
if there's any accessible stud at all - remove the pipe/cover/thermostat housing and then place a large nut on top of the remaining shank. weld the nut to the stubby shank, let it cool down, then use a socket on the nut. this method is awesome and quick/easy if you have a welder or known someone that does.
with enough shank left you may even be able to use vice grips or notch the shank and use a quality screw driver - if you have an impact screw driver that would be awesome. you smack them with a hammer and they twist a very small amount when hit - so it's like pushing in really hard to bite while backing the screw out. fantastic little gizmos.
if you have drill bits, drill the bolt out. small bit first, pilot hole, then larger. left handed bits are best as they may grab and back the bolt out but most folks don't buy those....unless they live in the rust belt and have to do this all the time.
if your drill bits suck really bad you can even drill the aluminum manifold to the side of the bolt (side away from the coolant passages) until you've got room to start knocking/beating the thing around and it'll come out. use a small bit, multiple holes and work your way larger, hogging out the aluminum. when you're done make a repair:
1. chase the threads with a tap or bolt and see if another bolt will work.
2. make larger threads
3. thread repair inserts
4. nut/bolt combo if there's room
5. drill a completely new hole in the ear and use a smaller bolt
6. ignore the stud and just use vice grips to hold the thermostat housing together - LOL
another thing i've done before - drill and tap a hole right inside the existing bolt shank that sheared off in the hole. i've done it to 12mm ABS sensor bolts, works fine on stuff that's non structural, doesn't need much more than just to hold it in place. i generally don't do this but sometimes you don't have many chioces and in this case it's such a benign and easy to seal part without any known issues there's plenty to work with.
there you go, about 10 options, all of whilch i've done before (except #6). there's a way to fix it. post a picture and someone will give you some specific tips.