Do you have a box grater? If so, here's what I found is best for handling cheese. Note that I'm using it for a 10 lb dog, and using a Microplane Box grater which has slightly wider holes than most other graters that I've owned.
grate cheese on the normal grating hole (the one that is 1/4 to 1/2" wide, depending on your grater). Spread the cheese out in a thin layer on a plate or cutting board and put it in the fridge for 1-2 hours until it feels dry to the touch but is still flexible. Still easy to break up, doesn't crumble, not greasy, and doesn't get mushed together, and is totally handleable after several hours at room temperature.
grate cheese on the normal grating hole (the one that is 1/4 to 1/2" wide, depending on your grater). Spread the cheese out in a thin layer on a plate or cutting board and put it in the fridge for 1-2 hours until it feels dry to the touch but is still flexible. Still easy to break up, doesn't crumble, not greasy, and doesn't get mushed together, and is totally handleable after several hours at room temperature.