Butter is great for general cooking but not frying. What is burning is the milk solids still left in the butter. Browning butter (basically lightly frying the milk solids) is a process that can be used to impart at different flavor to the butter.
If you want to fry with that buttery flavor but don’t want to burn it, you need to clarify it first: boil out the remaining water content, skim the milk solids off the top, then pour the clear fat off the other milk solids that settle out. Or you can just use ghee, which is basically preclarified butter.
Fun fact: it’s the milk solids that go off first in butter (and the water content allows microbial growth). “Potting” is a preservation technique like “canning”, except instead of sealing something with a metal lid, you pour clarified butter over the food. It fills in gaps between solid pieces and forms a cap on top as it cools.