How long after verbal offer do I wait for an official offer

Verbally offered the job on Monday after two rounds of interviews. I got the interviews through a referral, and so the head of the department I'll be working in said to submit a "formal application" through a link to a job posting he just created, and to let him know when I do so he can get me hired "asap".

I've used that platform to apply to another company before and it notifies you when your application has been viewed. He hasn't viewed mine.

I know that since it's a government organization things probably progress slower, but I'm a little impatient lol. Do you think it's wise to send an email asking for an estimate of the timeline to get an official offer? Should I wait till Monday to send that? Or not at all?

I want to give a two weeks notice to my current job but really do not want to be placed on another project after the one I'm working on ends (and it ends in two weeks)