I want to. Initially I was going to wait until the first batch was completely finished and ready to drink before I started my next one, only because it was my first time and I wanted to see how it turned out. I don't think I can wait another 3 weeks minimum before starting my next batch.
Make more beer now... it's called building up a pipeline. Having beers at different stages of fermentation and conditioning means that you won't run out of beer for one thing, also that you'll be able to let batches condition longer which means they'll have plenty of time to improve.
You'll also get more benefit from by actually improving your PROCESS by brewing in these next few weeks than you will for sitting around for 3 weeks waiting... PLUS you'll be less tempted to "sneak" a few to "check the carbonation level"
if you're actively engaged with other beers.
If you only have one fermenter and one secondary, AND you're choosing to move to a secondary rather than extended primary like many of us are doing.... you have the perfect opportunity to have 2 more beers ready to go by the time you've drunk this batch...
Brew, then in 2 weeks move that batch to secondary, and brew another batch in primary.
And then when you're bottling your second batch you can be enjoying your well conditioned first batch of beer.
If you're worried that this one might not turn out or whatever and that's why you're delaying... your beer will turn out FINE... it's really hard to screw up a batch... Look at what you did... you think you screwed up and we're telling you it will be fine, because we know from experience it will....
There's no logical reason to not brew in the interim, except your fear.....
And this isn't a hobby about fear... it ain't rocket science....it's one of the oldest and easiest processes on this planet for making a consumable product. It's been done since cavemen times with a lot less sophistication and know how (and things like sanitizers) than we have now...
The only way we usually ruin our beers is trying to "fix" a problem that isn't there... panicking that the airlock is not bubbling and messing with it... stuff like that.
Read
this article and get brewing again, you'll learn more in the next 3 weeks by brewing than you will by not brewing and thinking you're going to learn MORE by drinking your beer in three weeks....