How do I move onto other projects within my team or company?

My recommendation is to look at adjacent teams (often called "sister teams"). These are teams where they report to the same Director of Engineering that you do.

In general, you don't want to completely switch teams (i.e. change companies or go to a completely different VP) unless you absolutely have to, especially as a Senior/Staff engineer. The relationships and understanding of org culture are super important. This is why the most graceful move is to go to a sister team working on areas that you're interested in, brokering that transition with your manager and their team (and if it's a sister team, these 2 managers will probably know each other).

Senior engineers will often have good relationships with sister teams as well due to the cross-functional nature of your work, so you won't be starting from 0.

All that being said, you absolutely need to talk with your manager to figure out a move like this. These career transitions are so, so, so much harder when your manager is on your side vs. not. If you have at least a decent relationship with your manager, I highly recommend bringing this up in your next 1 on 1. It will feel awkward, but that's actually what 1 on 1 meetings are supposed to feel like - It's sort of the point 😉

Here's another great thread on how to pivot to a new tech domain: "How to transition from back-end development to distributed systems?"

Another thought is if there is a joint collaboration opportunity on your current team for a project that deals with cloud or security.

Perhaps you can borrow an idea or project from one of those teams and see how you could apply it onto your team.