There is nothing wrong with starting the sentence with "got". What's happening here is just that the subject of the sentence has been omitted. That is unusual in English, especially in formal and/or written English, but it happens rather often in spoken English. In this example, "got" is synonymous with "have". I understand that the verb "get" must be very confusing to people learning English, because we use it too much, and it has too many meanings. Unfortunately, nothing can be done about that. All I can say is that here it means "have". More specifically, since it is conjugated as "got" it could mean either "have" or "had", present tense or past tense, depending on context clues. Without any surrounding context, I think the implied subject is "I". Thus, the meaning of "Got a friend..." is "I have a friend...". However, if this sentence occurred in a context where a person was talking about someone else, then the implied subject would be the person already under discussion.

Starting a sentence with "Got" is improper English. It is absolutely unnecessary to have it in the sentence. I wouldn't call it an inversion sentence, it is just improper and makes someone sound uneducated. As far as what it means in this sentence, "I have" a friend of my sister that came across this solid glass piece. This means the same thing. The proper way to write this statement would simply be, "A friend of my sister came across this solid glass piece."

The puzzling thing in your sentence is not "got", but the phrase "came across this glass piece". What does that mean? Without understanding what that means, I cannot interpret what "got a friend" means.