I did journaling and sharing of my journals with two of my long term T's. Sometimes it was as much as 15-20 single spaced pages every week. Both of them did not bring up content from my journals unless I raised it first, which I think is different than what you are proposing, which is a written piece about something specific. It sounds like it was really helpful for you to let go of the piece and know she would read it, and I think it will be up to you to raise it at the next session. I would not assume that she will come prepared with notes like a creative writing teacher, or will give you any feedback on it unless you ask specifically.

I would also say that although the writing may feel to you like it speaks greatly to your experience, that's because you actually experienced it. It is quite another thing to write about a personal experience where other people get what you're trying to say. That's why not all memoirs are published and many that are, don't sell well. Same with personal essays based on experience. Maybe this applies only to me and most people consider me to be a good writer, but what I wrote and gave to my T's wasn't high on the communication scale. They didn't magically understand my experience or struggles just because I wrote them down. I had to talk about these things in therapy, to interact with them, in order for it to make sense.

With respect to darkside8 who mentioned the T referring to you "exact words," I'm not sure that is a realistic thing to expect. Because people use the same words and have very different meanings associated with them. Peruse a thread here that's been closed and see how this can operate even when people are making a big effort to try to understand and be understood.

I think therapeutic content is very difficult to achieve via the written word, and that's why T's are trained to interact and have sessions in person. Communication is at the heart of therapy, in the moment and sustained/linked over time. Not writing, not texts, not emails, etc. I'm not claiming these are never helpful but I think most of us who write things down assume that people grasp our meaning, and I think that happens a lot less often than we think. Writing to communicate a very personal, emotional experience is just extremely difficult, when your audience is someone other than you.