There was a patch of the cow tongue variety on the side of the house that I hated. Dug it out a week ago. I figured winter was a good time since the root system is weaker and shriveled. Also thought the spines on the dead stuff would have gone soft and I could grab it with my hands. WRONG... they were soft alright, and all over the palm of my hand. Had to take a break and get the tweezers. Wasn't near as painful as fresh sturdy spines though.
I used a ... tool... LOL... I don't know what it's called! It's a 3 prong rake of some kind. Grabbed it and pulled, raking it towards me, exposing the roots. Used a snow shovel to scoop it up. Regular shovel to get the roots. I DID NOT want it growing back.
But a whole pasture... I don't know about that one. Till the whole thing up and start over? (and not walking barefoot in it for a long time?) I don't like using chemicals, not sure if there is one for cactus.
What kind are they? How do they grow, seed or root spreading? I don't know much about cactus. But I know you want to go for them in their growing off season, and like any other seeding plant you hate, not when the seeds are ready for you to shake loose and start your problem all over again. If they spread by roots, poison or digging them out.
To be thorough it seems you'd have to destroy the whole area and start over, depends on how thick they are.
You could wait until they start actively growing then spray them with something like RoundUp.
Or do what I do which is to pull them up with a potato rake. Never have to touch them with my hands that way. Dump them into the bucket or wheelbarrow then use them to fill holes where stuff has dug under the fence.
I took out a section of cactus that was 30'x10' and several smaller patches. It was a lot of work, but not that horrible.
I took a bunch of beat up old trash cans and plastic bins, a machete and my bbq tools. Ours were the kind that you make cactus salad with (can't think of the name) I would hack off a chunk, use the bbq tongs or fork and put in the plastic bin. If you let them sit a bit, they dry out and are not heavy to dump. They go at $17 for 500 pounds here and you would be surprised at how heavy they can be!
Make sure you get all the green bits picked up. We have not had it come back. It took a while, but not the end of the world.
Those prickly pears we have here in Clay county are a pain and unfortunately they have to be dug up, take as much of the root system as you can or they will come back 2fold!!! Good luck on your search and we missed you at Bent Creek.....
There are specific herbicides for Prickly Pear or cactus, I cant remember the brand name but you do have to have a license for it though. If you dont want to use any chemicals......get a pitchfork and a grubbing hoe and some gasoline. First, use the pitchfork to get all the leaves off of that particular plant and make a pile. Second, use the grubbing hoe to dig out the root of the cactus or otherwise it will come back(rather quickly actually). Third, put all the roots, leaves etc in a big pile....let them dry out, soak em in gasoline and light em up. Those two are the only ways I know how to get rid of them. Well, I guess the other is pear burning with a blow torch to burn off all of the spikes so cows or goats will eat them.....
Hope it helps you out....