The most autonomy I've experienced at a big company, and the best work I've done, is when a small cross-functional team is told "go build this" and then left alone. The small team is allowed to come up with their own processes and management is only used for consultation on how to remove external obstacles. Many companies know this is highly effective because they'll set up such teams when a very important initiative comes up.
"Many companies know this is highly effective because they'll set up such teams when a very important initiative comes up."
I have been on a few such teams. The funny thing is that after the project has finished management will usually pad themselves on the shoulders, talk about the value of focused teams and immediately disband the team and go back to micromanagement mode.