If it is that tall and grown up from not being grazed in a year or 2, mowing will improve the rate the earthworms can incorporate the residual back into the ground making topsoil...and open up the canopy for things like clover and such to sprout also.
The more you cut your grass, the more it grows if it gets some moisture.... and if you cut the weeds before they go to seed, then the grass gets a fighting chance...
Don't "make them" eat down to the dirt, but don't give them hay if there is decent growth....
@mwj has good advice... watch their condition... and rotating will do more for the ground than anything in the long run. If it gets dry, if there is not enough moisture to keep the grass "growing" then supplement with hay as
@4hfarms suggested... moving where you feed so it can enrich the soil and spread some seed if there is any in it. But they are a grazing animal... they need to rustle their own grub.. Thanks for the advice! I started moving around where I feed and am keeping up on the rotations. Hopefully by fall and next year I'll see the progress.