There are two reasons that the article would be used. For items you can see and touch, "a part" is a piece of an object. It is countable and could be replaced. For example, my car needs a new headlight (a part). In your example, the piece of the steak that isn't cooked properly isn't countable. It is also neutral, not emotional. The other reason that the article is used is when we are talking about something emotional or philosophical. That is why we would say "a part of you".

"Part of you always doubts what you are doing" is fine, and sounds same as "A part of ...". And "A part of my steak" is also fine. Maybe I'm not the right person to answer, because I prefer less lawyerly rules about English as there are so many variations, formal vs conversational, American vs The Rest, dictionaries, etc.

Technically, both phrases are valid with and without 'a'. But the trend of using 'a' in the first phrase, and not in the last, is mainly a matter of smooth flow and emphasis, as far as I see.

В обох реченнях можна або включити, або пропустити артикль- обидва варіанти правильні