So yes FIFO should apply on shares where you have a gain.
Change of broker has no bearing.
If it's like previous scheme's I've seen you may have been selling shares on the vesting date to cover USC/PRSI/Income tax aka 'Sell to cover', there shouldn't be any problem there.
If you held shares, and they have gained in value, then from Revenue's perspective when you go to sell, they will be assuming you're selling the oldest shares first, and it's on that basis you should be calculating your CGT liability. (If you've done so between Jan-Nov this year, then the deadline is 15th December to declare and pay the CGT)