You'd want to start with the United States Digital Service (USDS). They're basically a consultancy inside the executive branch to build tech solutions for the various 3 letter agencies. You basically do a "tour of duty" for a year or so (however long you want), tackling one of these solutions. And they go out of their way to hire folks like you, from FAANG and other tech companies, so you work with good people.
https://www.usds.gov/