My pastures at the new place were covered in buttercups when we arrived this past Spring.
They are still there a little now but mostly gone already.
All I did was mow them down before the flowers went to seed. Let the grasses grow high for the Summer… mixing every time it got about knee high. Which took forever for most of the Summer since the grasses were so weak from overgrazing.
At the end of the Summer I let the grass grow high one more time and left it.
There is very little buttercup left out there now.
I will soon cut that grass short, and put it in the raised beds. Then heavily overseed the pasture with ruminant grasses and legumes.
Wish I had done that last Fall… I did in two patches and the new grass is thriving and green.
I will continue the tall/mow cycle until mid Summer again. By then we may get an animal or two to start light grazing. May wait until next year though if the grass is not matured yet.
I live in a heavily rained area though. This would take longer somewhere drier.
Edit: I just finished doing a bit of research. It depends on what type of Buttercup you have. I think I may have one of the annual variants. Which is why growing the grass tall and thick by the end of Summer seems to have kept it from germinating much in the field. It is a Winter annual… that blooms in the Spring.
https://youtu.be/rIucB8oKVys
https://youtu.be/DImCyJBn674