Same subnet, two routers

I am not exactly a networking person, but I once solved a networking problem in my company and apparently now I am the guy that people ask stuff when they have network problems. I am not complaining, I am eager to learn but sometimes I just don't know how to solve a problem.
So here's the problem my colleague presented to me today:

-She has a subnet, let's say 10.10.10.0/24, on geographically two different locations and these can't be changed. These two routers are directly connected with a cable. No routing over the internet. Both locations have switches behind the routers, both locations have VLAN10.

- Apparently she wants these two locations to be in the same LAN. So let's say we have PC-A in siteA with an address of 10.10.10.10 and PC-B at siteB with an address of 10.10.10.20.

​

So the first question is that is this sorta thing even possible? As far as I know, a Switch is a broadcast domain, and again a VLAN is also a broadcast domain of its' own. Does Layer 2 connectivity limit to a broadcast domain? If it does, wouldn't this basically mean that I cannot stretch this LAN across two routers and I would have to do routing with this case? Would pc-A have connectivity with pc-B without routing as the two routers are directly connected?

Sorry , this is a newbie question and I mightve phrased it confusingly. Any help would be appreciated! Thanks.