I'm no expert on grazing but during my geology studies i was taught that a lot of the prehistoric climate shift, soil degradation and subsequent erosion around the mediterrean was partially caused by logging and subsequent overgrazing by goats and sheep which prevented spontaneous regeneration of forests.
Whatever critter you use - the taste might be wonderfull
One way to cope with erosion is by terracing your hill.
In prehistoric times till the early 20th century, farmers in Europe did this (among other methods) by a combination of hedgerows and plowing. Essentially, you establish a hedgerow more or less around a contour line on your property. Then you plough the lower part so that the furrow falls downhill and the upper part falls the other way.
The hedgerow breaks the speed of the runoff and slows down and catches any eroded material from uphill. So the lower part of the uphill part rises while the top of the plot loses height due to erosion and plowing. Each plot on your hill side experiences a reduced soil loss. Ultimately you get a short, steep hillside under the hedge followed by a flattened area untill you reach the next hedge.
We call this landscape form a 'graft'. There is a wiki that describes more or less how it looks but the way it forms could be improved upon. It's Dutch so use autotranslate.
https://nl.wikipedia.org/wiki/Graft_(hellingknik).
The idea to 'build' fields of this nature has some comparisons with the english ridge and furrow system
https://en.wikipedia.org/wiki/Ridge_and_furrow The closest english wiki to a graft is the lynchet landscape
https://en.wikipedia.org/wiki/Lynchet You could consider a graft as an improved version of the lynchet where the hedgerow catches runoff and the transported sediment. The hedgerow also provides an improved microclimate.
Another way was by actually building terraces. This was really labour intensive and was only done to support cash crops (wine) or near settlements where space was at a premium. Check pictures of old medieval terracing.