Stay on big tech for management experience, or move to startup?

Hello hacker news, I'm facing a conundrum in my career, I'm a Senior Software Engineer at a Big Tech company, and I'm very bored at my job, I'm using boring tech, things move slowly, and I have basically no power to change any processes, so I definitively want to move from there, but I have two good opportunities to decide on: stay on the Big Tech, but move to an Engineering Manager position on a newer and growing part of the company with less headcount OR move to a new but promising Startup, which has been doubling in size, as Principal Developer of whatever (not really important because it should be easy to move inside)

Going forward, I would really like to put my soft skills to practice, to improve processes, to change things for the better, because I feel I haven't been doing that for too long, of course I think I would be able to do way more of that on the startup, but I also think I would be able to have some of it on this big tech position

So then the main puzzling part for me then is, if I join the startup now, I will probably be able to drive much of its architecture, have more fun, and grab more equity and position early on, and I don't want to miss this opportunity. On the other hand, I don't have any hard management experience in my resume, staying my employer will give me that with a good name, so if I move a a startup after that and later want to go back to any big tech as an engineering manager I'll be able to, and I would very much like to keep this optionality

Help me?