How to be direct and compassionate while giving feedback AND maintain staff’s respect?

Hi! I’m in my second year in a leadership position at a private school. For the most part, I feel like I have been doing well and growing my professional and leadership skills. However, many things remain challenging. I think I am respected for my skills, knowledge, and “calm” demeanor. I generally am seen as being kind and I have a very soft speaking voice. I also went from being “peers” with the majority of my staff to suddenly being the director of their department, which is a tricky dynamic.

Recently, I was involved in a conflict resolution meeting with one of my staff members. I can see both sides of the argument and after providing what I felt was validation for her negative experience, I wanted to give her feedback. It did not go well. She had somewhat of an emotional breakdown when I pointed out things that I thought were not professional and would reduce her impact/credibility. She said she thought of us as friends and thought it was a safe space but now knows better. She said that I am not supporting her. Another staff member got involved (hearing her side of the story only and said I was being too hard, unfair, and unsupportive).

This, combined with a couple other incidents makes me feel like I am not expected to give ANY directive or corrective feedback and if I do, it’s characterized as a personal flaw of mine.

I need to be able to give feedback but I also want to be compassionate. Help!