How can you cook milk without it burning the pan?

Double boiler.

You can buy a purpose built double boiler and if you are going to do this often you should. But for now you can fake it the way I do. Here is my setup.

The little saucepan is floating in the water that is in the larger pan. Because of the water, the temperature never gets higher than 100°C unless you boil all the water off. Milk at 100°C will boil but it will not scorch. That is milk gravy (white sauce) in the small pan which happened to be in there and which I thought was a good illustration.

Float a small pan in a larger pot of water and cook your milk in the small pan. If you like it, buy a double boiler. Sometimes the floating pan is too heavy and some water comes in over the side. Not a big deal for milk but a total loss if you are using the setup to melt chocolate chips.

There are a number of ways to cook milk without burning it.

My personal favorite is to bake it. When baked according to my instructions, it does not overflow, and the bottom of the pan does not burn at all (the edge where the milk surface touches does get some milk protein cooked onto it, though). You do not have to do anything with it while it's baking. It does develop a significant brown skin on top, but you can pull it off and eat it. To cook it to my personal satisfaction, I fill an 8"x8" glass pan about halfway with milk, and put it in the toaster oven on 425° to 450° F. or so for about 25 minutes. I like to make milk porridge via this method, too (same instructions, except add oats and brown sugar with the milk). Anyway, I tolerate it better than boiling on low heat on the stove, and it's much easier. 25 minutes on 450° F. might be overkill for some people's needs, especially if they don't need to kill anything in it, nor denature any components of it, but it's not overkill for a non-soupy milk oat porridge. It is possible to burn the top of the milk if you bake it too long/hot or too close to the surface of the oven—but you can just take the skin off, and your milk should taste fine. If you bake it in a toaster oven (on the bake setting), do not cover it, or it will not cook very quickly.

Here's what the milk looks like after it's been baked (in my toaster oven on bake on 450° F. for 25 minutes), with the skin. The skin is dark brown (although not black, even though parts of it look black in the picture; the lighting wasn't perfect, and it's just my inexpensive tablet's camera); so, I probably could have used a lower heat than 450° F. (like 425° F.), but this does not affect the taste of the milk underneath it anyway, since the top side of the skin does not touch the milk; I usually don't cook just plain milk with nothing in it, so that accounts for the error:

Here's the baked milk after removing and eating the skin (this remaining milk is really what we're after):

Here's the pan after I drank the milk (you can see that there isn't milk burned on the bottom of the pan, and it should be fairly easy to clean compared to a burned pan; there are cooked portions stuck where the milk surface was, but they're not burned); it tasted great, by the way (and not as much like evaporated milk as stovetop boiled milk):

Other methods:

Some people like to microwave milk. It can overflow easily that way, if you're not careful, however, but it shouldn't burn. For personal reasons, I don't recommend using a microwave for anything, though, but if you do, I don't blame you.

Boiling milk on the stovetop without burning it requires a lot of patience, skill, and attention. People like to use pans with thick bottoms, since they heat more slowly, and are less prone to making things burn (not just milk). They like to heat it at a low temperature. They recommend stirring. In my experience, it can still burn even then, though, if you cook it long enough.

Some people recommend putting a very small amount of cold water on the bottom of the pan when boiling milk to help prevent it from burning. I'm very skeptical about the logic entailed, since the milk is cold anyway (and the water at the bottom of the pan will mix with the milk, especially if you stir it as I saw recommended with this method; so, it will no longer be on the bottom of the pan), but if it works, it works, and I don't claim whether it works.