Many of us have come across studs broken off...excuse the terminal preposition. Here's an example:
Let's say you were changing the O2 sensor...an MK1 is shown below:

Next you find that one of those little bolts has been broken "off" inside the exhaust manifold:

So what should you do?
Most guys just put it back with two bolts, or leave it off.
Some people have claimed that the "easy out" is the way to go. But I've tried many times and had no luck with an 'Easy Out'...
Some need expensive tools, or will send it to a machine shop.
Others weld something on, and wrench on that.
But what if it's flush, or recessed a bit?
Finally, an example showed up, so I thought I'd run through a quick, easy, and safe way to do it yourself.
First, drill a pilot hole into the center of the broken stud:

Now that you've got a good start, continue all the way through the old bolt. Here, I've placed the correct bolt next to the pilot hole, so you can see how deep to drill.

Some people get scared that they can't hit the center of the bolt. Others are worried that they'll mess up the threads. Just using a hand held drill, and with the exhaust manifold on an upside down trash can, it wasn't too hard for me to get it pretty close to center. No big deal if you take your time. No big deal if you aren't perfectly centered.
Now...we're into it about 7 minutes. Here's a blurry picture of what the hole looks like as I change to a larger bit. Notice that the bolt is getting pushed into the threads of the exhaust manifold. There is no danger to the threads.

Again, a few minutes later, the second drill bit has passed through the bolt. Although a little blurry, you can still see lots of metal between the drill and the threads.

Now, we pass still another drill bit through the hole, this time rendering the old, sheared off bolt paper thin. This allows us to put a tap through the threads, and that breaks up the last of the bolt. It's slow going at first, but once you get the tap started, it's a piece of cake to turn it in a few turns, and then clean out the pieces, and so on, until you end up with this:

Just in case you are confused about what kind of tap I used, here's a picture of it being used to clean up the threads on the lower part of the exhaust manifold:

The time to do this entire 'job' was less than 30 minutes.
I've read well meaning posts which have side-stepped or 'poo-pooed' this method as being too tough for the average individual. The tools I used here were ordinary tools that are commonly available.
You'll never know how easy it is until you actually
try it.