There is no [][] operator. What actually happens is that the second [] operates on the variable returned by the first []. Because there is already that functionality, it would create ambiguity were there to exist a [][] operator.
For example: let's say you have a variable x of some type T.
T x = new T();
If we use the [] operator, let's say a variable of other type Q is returned:
Q y = x[0];
And then using the [] operator on a variable of type Q might return a variable of type R:
R z = y[0];
Therefore x[][] returns a variable of t ype R.
Let's say we actually were able to overload [][] for type T such that it returned a type S:
S a = x[0][0];
The compiler would have no way of knowing if it should use the [][] operator on x to return a type S variable, or use the [] operator twice in a row to return a type R variable. This is the ambiguity I mentioned above.
Your best bet if you're stuck on using square brackets is to have operator[] return a variable which also has [] overloaded (or perhaps a variable of the same type, with a flag set), and have that initially returned variable deal with the second [].
But the best solution here (as mentioned already in another answer) is to use a different operator such as ().