You’re the first one I’ve read that makes it like me to include carrots!
I come from Hartlepool in County Durham (I now live in the south east) where we call it Panackelty, though. I understood that Panhaggerty is all about cheese rather than corned beef and hails from a bit further up in Newcastle.
My mam (and now me) slice up corned beef from a can, white onions, potatoes AND carrots. My mam always did it in a large deep frying pan on the top of the oven but I mostly put it in the oven in a dish of equal depth but I have also made it in a matter of 10/12 mins in a pressure cooker, many times too.
Cover the completed layers (no particular order, just keep layering until you run out) with a dissolved beef OXO cube in a pint of water until it becomes visible through the top layer; be careful not to add too much water or you’ll end up with soup.
Use your preferred method and cook or simmer until potatoes are tender. Potatoes should be just less than a cm depth or you’ll be starving before you sit down.
One tip I’ll give you for an even tastier version (not in any way authentic but enhances it in my humble opinion) is to add a teaspoon of mild curry powder to the stock. It does the trick!