Viola, yes, "but" can definitely start a sentence. In fact, some of the best writers do so.
BUT learners are advised NOT to do so until they are advanced users of the language.
Some learners start too many sentences with "but," so it becomes absurd.
"But" is often placed at the beginning of a sentence for effect (drama).
I have made up this example. It is NOT very good. But maybe it gives you an idea.
"I love doughnuts. There's a doughnut shop on my way home from work. I often think about going in and buying a dozen. BUT I don't, (More dramatic or emphatic than "I often think about going in and buying a dozen, but I don't.")
P.S. Most Americans now spell "doughnut" as "donut."