How do you a publish a book piecemeal online while building an audience?

After I finished the rough draft of my book, I started pondering how to bring it to life. I could take it directly to Kickstarter and just see how it fared. I'd probably attract enough attention to publish the book on my own. But I wanted to get direct feedback on the book and build a group of people who felt that the contents of the book resonated with them. This, to me, seemed like the only way to make the book as good as it could possibly be.

I had already ruled out Medium and other common writing and blogging sites, as they didn't quite do what I wanted (although I will likely cross-post some chapters to raise interest in the book).