The Halacha permits or chopping or cutting cheeses into small pieces on Shabbat. The prohibition of Tohen does not apply to cheese, since it only applies to “Giduleh Karaka” things that grow from the ground. However, Maran (Siman 321) adds one stipulation: One may not cut the cheese with a grater or garlic with a special smasher, normally used to grate things into small pieces. This is forbidden on Shabbat as it is a weekday type of apparatus used for grinding.
However, it is permitted to use a knife or a special cheese knife. Similarly, it is permissible to smash tuna fish or eggs with a fork, as they do not grow from the ground. In all these cases, not only is it permitted to chop or smash into very small pieces, but it may be done well in advance of consumption. For example, one may smash the hard-boiled egg on Shabbat morning for Seuda Shlishit.
It is permitted to chop cheese, eggs and fish on Shabbat, as long as a grater is not used.