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.