How do I approach a supervisor about an open-ended PhD project listing?

Hi everyone,
I am an MSc graduate and was recently looking at PhD opportunities. I found a PhD listing that really interests me, but I’m not sure how to best reach out to the supervisor. It’s not a fixed project - instead, the description outlines a broad theme around adjustment, coping, and well-being in long-term neurological conditions. The scope is wide, covering things like mindfulness, CBT, experimental cognitive training, neuroimaging, and even app development. The advert invites potential candidates to email the supervisor to discuss their interests in relation to the project.

Because I am new to PhD applications and interested in these aspects, hence I wanted to ask for some guidance. When first contacting the supervisor about this kind of open-ended PhD project, is it better to briefly introduce my background and general interests, or should I already propose a rough project idea (e.g., a specific intervention or method)?

Any advice on how to make a good first impression and not overwhelm the supervisor with too much (or too little) detail would be really appreciated!