I'm always surprised when I hear of guys going out and breaking up cowpies... and I'm equally always surprised at how I see them disappear almost magically when the ground thaws out and the soil biology gets revved up, all on their own.
I'm not judging anyone when I say that... but my attitude has always been to just leave them. True, I generally don't make hay as a general rule on my pastures where I winter AND summer my cattle. I feed them by unrolling hay right on the same pastures they're on all the time... they never leave them, other than through normal rotation. In winter, they have free range of all of them, and I do whatever rotation I intend by changing where I feed them. But in areas like behind a grove or where I want to do some really heavy application to improve the poor soil, etc., the cowpies are pretty much touching each other, and in winter, it's a real pain to drive over those 5" high piles of crap!
Within a month of thaw up, the ground is smooth, and the grass is growing right on up through, and ultimately, you can't even tell that it's the same spot anymore. IF I were to cut hay on some of these areas (which I do) I just cut slightly high (a good practice anyway), and I don't get any manure in the hay. I also then can rake a bit high (I'm using a basket rake, NOT a wheel rake...), and the hay is clean. After many years of watching those cowpies disappear, I just see no reason to go out and drive in circles...
But perhaps in your conditions, it's necessary...