The website attached to the Facebook pages is Americancolumn.com
https://www.facebook.com/Proud-Ameri...4162400004601/
https://www.facebook.com/AmericanColumn/
It has two pages, and started both in November/December of 2015. How is this even possible?
I have a page growing at a steady 15-20k per week, but 400k in a month is unheard of...
Not impossible given how hot under the collar so many Americans are right now about gun laws and all that ridiculous stuff.
Chances are, they just accidentally hit the zeitgeist. It happens from time to time - you write or create something and it takes off beyond all expectation.
You can't really force or plan that, although there are various techniques to improve your chances (take risks with your content, be divisive, appeal to core values). But if you do enough stuff it'll happen eventually.
Hm, I see what you're saying, but I've just seen it happen far too often with these political pages to think that it's just luck. Pretty much all of the memes they share blow up and go viral.
Maybe some kind of blackhat software?
it doesnt look right to me
when you click on the "Like amount" in the page it shows you how much likes the page got on the last days and in both pages it shows 0
doesnt look like 380K + likes and nothing came in the past week or so when taking into consideration this started in November/December of 2015

Could be blackhat in some way - not my area of specialisation, but given the evidence, yup, looks suspect.
check out these pages you might get a better understanding how they managed to do it
http://likefake.com/
http://www.tricksage.com/2013/10/get...likes-for.html
https://www.youtube.com/watch?v=s7vG4wazhoA
https://www.youtube.com/watch?v=tp3g8CaF30k
and there are so many more 
Yeah no likes at all in the past week = definitely suspect, probably all fake followers and reposts/tagging to grow apparent activity.
There will be ones like this that are legit, just not this one.
The activity of the page isn't that good for 350k followers, so maybe a bunch of them are fake. But not all of them cause the fan page isn't dead either.
http://kingdomlikes.com/
^
Not an aff link + free
There's value in *appearing* big. I've tested results of youtube view count/likes, twitter followers, etc and assure you it makes a BIG difference. 95% people don't spend the time to inspect whether the stats are legit (probably 99% plus, honestly).
The first link affiliaxeguy posted is hilarious. Those are the guys who made the "Drake shake" app. I got this banner idea from them:
(The winning the lottery vs banner clicks is statistically true)
Easy, look at the engagement on posts ... scroll down to 24 hours or so, if they have just a few likes, shares and a few comments, then.... probably a lot of fake likes... a fan page with a lot of engagement shows fans are real generally.
They have a fair amount of engagement - looks like a bunch of real followers with their numbers buffed up with fakes.
SimilarWeb claims they're getting 50% of their website traffic from their Facebook page:
http://www.similarweb.com/website/am...umn.com#social
... of course, it could just be estimating those numbers based off the Facebook activity.
Their social strategy is pretty ironclad, however; they're posting new content every 2 hours like clockwork, and it is consistently polarizing, inflaming content that whips people up and makes them want to comment and share.
One of the best ways to get lots of people sharing: publish something controversial, offensive, or that "exposes" things that helps sharers confirm their darkest suspicions.
Seems like all these Facebook pages are doing is that, over and over. Proven recipe for social virality.
Chase
Here's an interesting page: http://www.socialbakers.com/statisti...6728-facebook#
Maybe you'll find some interesting numbers 
We've had pages go from 1 to 2 million within a week. It's very possible.